Graphing linear equations is a fundamental skill in mathematics, providing insight into the relationship between variables. Among these equations, y = 1/2x holds a unique place with its simplicity and prevalence. Whether you’re a student encountering it for the first time or an experienced grapher seeking a refresher, understanding how to graph y = 1/2x is essential. This article will guide you through the process, shedding light on its properties and providing a step-by-step approach to visualizing this equation.

The graph of y = 1/2x is a straight line that passes through the origin (0, 0). Unlike other lines with an intercept, this line has a slope of 1/2, which determines its steepness and direction. Positive values of x will produce positive values of y, resulting in a line that rises as you move from left to right. Conversely, negative values of x will produce negative values of y, creating a line that slopes downward. This distinct behavior distinguishes y = 1/2x from other linear equations and highlights its unique characteristics.

To plot this line, follow these steps. First, locate the y-intercept, which is (0, 0) in this case. This point represents the intersection of the line with the y-axis. From there, use the slope to determine the direction and steepness of the line. With a slope of 1/2, move up 1 unit and right 2 units from the y-intercept. This gives you another point on the line, (2, 1). Repeat this process until you have several points, and then connect them to form the line. The resulting graph will be a straight line passing through the origin and exhibiting the characteristic slope of 1/2.

Understanding the Equation of a Parabola

A parabola is a U-shaped curve that is symmetrical about a vertical axis. The equation of a parabola can be written in the form
$$ y = ax^2 + bx + c $$
where a, b, and c are constants.

The value of a determines the shape of the parabola. If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. The absolute value of a determines the steepness of the parabola. A larger value of a will result in a steeper parabola.

The value of b determines the horizontal shift of the parabola. If b is positive, the parabola is shifted to the left. If b is negative, the parabola is shifted to the right. The value of b is equal to -2 times the x-coordinate of the vertex of the parabola.

The value of c determines the vertical shift of the parabola. If c is positive, the parabola is shifted up. If c is negative, the parabola is shifted down. The value of c is equal to the y-coordinate of the vertex of the parabola.

To graph a parabola, you can use the following steps:

1. Find the vertex of the parabola. The vertex is the point where the parabola changes direction.

2. Plot the vertex on the coordinate plane.

3. Find two other points on the parabola.

4. Draw a smooth curve through the three points.

The following table summarizes the key features of a parabola:

Feature	Equation
Shape	a
Horizontal shift	b
Vertical shift	c
Vertex	(-b/2a, c – b²/4a)

Identifying the Vertex of the Parabola

Identifying the vertex of a parabola is a crucial step in graphing it accurately. To determine the vertex, you need to use the formula x = -b/2a. Let’s apply this formula to the equation y = 1/2x:

**Step 1: Identify the values of a and b.** In this case, a = 1/2 and b = 0.
**Step 2: Calculate x:** Using the formula x = -b/2a, we have x = -(0)/2(1/2) = 0.
**Step 3: Calculate y:** To find the y-coordinate of the vertex, plug the x-value into the original equation: y = 1/2(0) = 0.

Therefore, the vertex of the parabola y = 1/2x is (0, 0).

Additional Details:

The vertex of a parabola represents the point where the parabola changes direction. It is the lowest or highest point on the graph, depending on whether the parabola opens up or down, respectively.

The formula for finding the vertex of a parabola in the form y = ax^2 + bx + c is:
“`
x = -b/2a
y = f(x) = a(x^2) + bx + c
“`

Here’s a summary of the process in tabular form:

Step	Formula
Identify a and b	a = coefficient of x^2 b = coefficient of x
Calculate x-coordinate	x = -b/2a
Calculate y-coordinate	y = f(x)

Understanding the vertex of a parabola is essential for graphing it accurately and analyzing its key characteristics.

Determining the Axis of Symmetry

In mathematics, the axis of symmetry is a vertical line that divides a graph into two symmetrical halves. For the equation y = a(x – h)^2 + k, the axis of symmetry is x = h. To determine the axis of symmetry for the equation y = 1/2x, we need to convert it into the standard form y = a(x – h)^2 + k.

Step 1: Multiply both sides of the equation by 2 to get rid of the fraction.

“`
2(y) = 2(1/2x)
2y = x
“`

Step 2: Rewrite the equation in the standard form y = ax^2 + bx + c.

“`
2y = x
2y – x = 0
“`

Step 3: Convert the equation to the standard form y = a(x – h)^2 + k.

“`
2y – x = 0
2y = x
y = 1/2x
“`

Since the equation is already in the standard form, we can see that the axis of symmetry is x = 0.

Additional Information

The axis of symmetry is an important property of a parabola. It can be used to find the vertex, focus, and directrix of the parabola. The vertex is the point where the parabola changes direction. The focus is the point that the parabola is focused on. The directrix is a line that is perpendicular to the axis of symmetry and passes through the focus.

The following table summarizes the key properties of a parabola:

Property	Formula
Vertex	(h, k)
Focus	(h + p, k)
Directrix	x = h – p
Axis of Symmetry	x = h

where p is the distance from the vertex to the focus or directrix.

Finding the Intercepts

The intercepts of a linear equation are the points where the graph crosses the x-axis and y-axis.

To find the x-intercept, set y = 0 and solve for x.

y – 1 = 2x

0 – 1 = 2x

-1 = 2x

x = -1/2

So, the x-intercept is (-1/2, 0).

To find the y-intercept, set x = 0 and solve for y.

y – 1 = 2(0)

y – 1 = 0

y = 1

So, the y-intercept is (0, 1).

The intercepts of the line y – 1 = 2x are (-1/2, 0) and (0, 1).

Plotting Points on the Parabola

To plot points on the parabola of the equation y = 1/2x², determine two $x$ values at which to calculate the corresponding $y$ values. One $x$ value should be positive, and the other should be negative. Here’s a step-by-step guide on plotting points:

Step 1: Choose $x$ values

Select two $x$ values, one positive and one negative. For instance, consider $x = 2$ and $x = -2$.

Step 2: Calculate $y$ values

For each $x$ value, substitute it into the equation $y = 1/2x²$ and calculate the corresponding $y$ value. For $x = 2$, we get $y = 1/2(2)² = 2$, and for $x = -2$, we get $y = 1/2(-2)² = 2$.

Step 3: Create a table of values

Organize the $x$ and $y$ values in a table. This table helps visualize the relationship between the $x$ and $y$ values.

$x$	$y$
2	2
-2	2

Step 4: Plot the points on a coordinate plane

Use the $x$ and $y$ values from the table to plot the points on a coordinate plane. The point (2, 2) is plotted in the first quadrant, and the point (-2, 2) is plotted in the third quadrant.

Step 5: Connect the points with a smooth curve

A parabola is a smooth, U-shaped curve. Once the points are plotted, connect them with a smooth curve. This curve represents the graph of the equation $y = 1/2x²$.

Drawing the Graph of Y = 1/2x

1. Intercept

The intercept is the point where the graph crosses the y-axis. To find the y-intercept of Y = 1/2x, we set x = 0 and solve for y:

y = 1/2(0)
y = 0

Therefore, the y-intercept of Y = 1/2x is (0, 0).

2. Slope

The slope of a line is a measure of its steepness. To find the slope of Y = 1/2x, we can use the formula:

slope = (change in y) / (change in x)

Let’s take two points on the line: (0, 0) and (1, 1/2). The change in y is:

change in y = 1/2 - 0 = 1/2

The change in x is:

change in x = 1 - 0 = 1

Therefore, the slope of Y = 1/2x is:

slope = 1/2

3. Plotting Points

To graph Y = 1/2x, we can plot a few points and connect them with a line. Here are a few points on the graph:

(0, 0)
(1, 1/2)
(-1, -1/2)

4. Drawing the Line

Once we have plotted a few points, we can connect them with a straight line. The line should pass through the origin and have a slope of 1/2.

5. Asymptotes

An asymptote is a line that a curve approaches but never touches. Y = 1/2x has two asymptotes:

Vertical asymptote: x = 0
Horizontal asymptote: y = 0

The vertical asymptote is a vertical line that the graph approaches as x gets closer and closer to 0. The horizontal asymptote is a horizontal line that the graph approaches as x gets larger and larger.

6. Vertical Line Test

The vertical line test is a way to determine if a graph represents a function. A graph represents a function if every vertical line intersects the graph at most once.

To perform the vertical line test on Y = 1/2x, we can draw a vertical line at any x-value. For example, let’s draw a vertical line at x = 1. The line intersects the graph at (1, 1/2). There is no other point on the graph that intersects the vertical line. Therefore, Y = 1/2x passes the vertical line test and represents a function.

Detailed Explanation of the Vertical Line Test

The vertical line test is based on the definition of a function. A function is a relation that assigns to each element of a set a unique element of another set. In other words, a function is a rule that assigns a unique output to each input.

The vertical line test is a way to determine if a graph represents a function because it tests whether every input has a unique output. If a vertical line intersects a graph at more than one point, then there is at least one input that has two different outputs. This means that the graph does not represent a function.

In the case of Y = 1/2x, every vertical line intersects the graph at most once. This means that every input has a unique output, and therefore Y = 1/2x represents a function.

The vertical line test is a useful tool for determining if a graph represents a function. It is a simple test that can be applied to any graph.

How to Graph y = 1/2x

Shifting the Graph Vertically: Adding or Subtracting a Constant

Adding a Constant

To shift the graph of y = 1/2x vertically upward by a units, add a to the right-hand side of the equation:

“`
y = 1/2x + a
“`

To shift the graph downward by a units, subtract a from the right-hand side of the equation:

“`
y = 1/2x – a
“`

Example

To graph the function y = 1/2x + 2, add 2 to each y-coordinate of the original graph. The result is a graph that is shifted 2 units upward.

| x | y = 1/2x | y = 1/2x + 2 |
|—|—|—|
| -2 | -1 | 1 |
| -1 | -1/2 | 1.5 |
| 0 | 0 | 2 |
| 1 | 1/2 | 2.5 |
| 2 | 1 | 3 |

Subtracting a Constant

To shift the graph of y = 1/2x vertically downward by a units, subtract a from the right-hand side of the equation:

“`
y = 1/2x – a
“`

To shift the graph upward by a units, add a to the right-hand side of the equation:

“`
y = 1/2x + a
“`

Example

To graph the function y = 1/2x – 1, subtract 1 from each y-coordinate of the original graph. The result is a graph that is shifted 1 unit downward.

| x | y = 1/2x | y = 1/2x – 1 |
|—|—|—|
| -2 | -1 | -2 |
| -1 | -1/2 | -1.5 |
| 0 | 0 | -1 |
| 1 | 1/2 | -0.5 |
| 2 | 1 | 0 |

Shifting the Graph Horizontally: Multiplying by a Constant

Multiplying the input of a function by a constant will shift the graph horizontally. For example, the graph of y = x² can be shifted 3 units to the left by multiplying the input by 3, resulting in the function y = (3x)². This shift is because the values of x are now 3 times smaller, which means that the graph will be stretched horizontally by a factor of 3.

In general, if f(x) is any function and c is a constant, then the function g(x) = f(cx) will be the graph of f(x) shifted c units to the right if c is positive and c units to the left if c is negative.

For example, the graph of y = x² can be shifted 2 units to the right by multiplying the input by 1/2, resulting in the function y = (1/2x)². This shift is because the values of x are now 2 times larger, which means that the graph will be compressed horizontally by a factor of 2.

The table below summarizes the effects of multiplying the input of a function by a constant:

Constant	Shift
c > 0	c units to the right
c < 0	c units to the left
c = 1	No shift

Multiplying the input of a function by a constant can be a useful tool for transforming graphs. For example, it can be used to shift a graph so that it passes through a specific point or to align it with another graph.

Here are some additional examples of how multiplying the input of a function by a constant can be used to shift its graph:

To shift the graph of y = x² up 2 units, multiply the input by 1/2, resulting in the function y = (1/2x)².
To shift the graph of y = x³ down 3 units, multiply the input by -1, resulting in the function y = (-x)³.
To shift the graph of y = sin(x) to the left by π/2 units, multiply the input by 2, resulting in the function y = sin(2x).

Multiplying the input of a function by a constant is a simple but powerful tool that can be used to transform graphs in a variety of ways.

Reflecting the Graph over the x-axis

When you reflect the graph of y = 1 – 2x over the x-axis, the resulting graph is y = -1 – 2x. This is because the y-coordinate of each point on the original graph is negated when the graph is reflected over the x-axis. For example, the point (1, -1) on the original graph becomes the point (1, 1) on the reflected graph.

To reflect the graph of y = 1 – 2x over the x-axis, you can use the following steps:

Plot the original graph of y = 1 – 2x.
For each point (x, y) on the original graph, find the corresponding point (-x, -y) on the reflected graph.
Plot the points from step 2 to create the reflected graph.

The following table shows the coordinates of the original graph and the reflected graph:

Original Graph	Reflected Graph
(1, -1)	(1, 1)
(2, -3)	(2, 3)
(-1, 3)	(-1, -3)

As you can see from the table, the y-coordinates of the points on the reflected graph are the negatives of the y-coordinates of the points on the original graph. This is because the graph is reflected over the x-axis.

Reflecting the graph of y = 1 – 2x over the x-axis is a simple transformation that can be performed in a few steps. By following the steps outlined above, you can easily create the reflected graph.

Examples of Graphing Y = 1/2x in Different Forms

Slope-Intercept Form

In slope-intercept form, an equation is written as y = mx + b, where m is the slope and b is the y-intercept. To graph y = 1/2x in slope-intercept form:

Plot the y-intercept: Start by plotting the y-intercept, which is 0. Mark a point at (0, 0).
Find another point: Choose any non-zero value for x and calculate the corresponding y-value using the equation y = 1/2x. For example, if you choose x = 2, then y = 1/2(2) = 1. So, mark a point at (2, 1).
Draw a straight line: Connect the two points you have plotted using a straight line. This line represents the graph of y = 1/2x in slope-intercept form.

Point-Slope Form

In point-slope form, an equation is written as y – y₁ = m(x – x₁), where (x₁, y₁) is a point on the line and m is the slope.

Choose a point: Select a point on the line, such as (2, 1) from the previous example.
Calculate the slope: The slope of the line is 1/2.
Write the equation: Substitute the point and slope into the point-slope form equation: y – 1 = 1/2(x – 2).

Intercept Form

Intercept form is written as y = a/b, where a is the y-intercept and b is the x-intercept. To graph y = 1/2x in intercept form:

Plot the y-intercept: The y-intercept is 0, so mark a point at (0, 0).
Find the x-intercept: To find the x-intercept, set y = 0 and solve for x: 0 = 1/2x => x = 0. So, the x-intercept is 0.
Draw a vertical line: The graph of y = 1/2x in intercept form is a vertical line passing through the y-intercept and the x-intercept.

Standard Form

Standard form is written as Ax + By + C = 0. To graph y = 1/2x in standard form, reorganize the equation to:

2x – y = 0

Using the standard form, we can determine the slopes and intercepts directly:

Slope: The coefficient of x, 2, represents the slope of the line.
x-intercept: The coefficient of y, -1, represents the x-intercept when y = 0: (0, -1).
y-intercept: The constant term, 0, represents the y-intercept when x = 0: (0, 0).

Two-Point Form

Two-point form is written as y – y₁ = (y₂ – y₁)/(x₂ – x₁) (x – x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.

Choose two points: Select two points on the line, such as (2, 1) and (4, 2) from the slope-intercept form example.
Substitute the points: Plug the points into the two-point form equation: y – 1 = (2 – 1)/(4 – 2) (x – 2).
Simplify: Simplify the equation to get: y – 1 = 1/2 (x – 2).

Table of Graphing Methods and Equations

Form	Equation	Plotting Steps
Slope-Intercept Form	y = 1/2x	– Plot y-intercept (0, 0) – Find another point (2, 1) – Draw a line
Point-Slope Form	y – 1 = 1/2(x – 2)	– Plot point (2, 1) – Use slope (1/2) to find direction
Intercept Form	y = 0	– Plot y-intercept (0, 0) – Draw a vertical line
Standard Form	2x – y = 0	– Find x-intercept (0, -1) – Find y-intercept (0, 0) – Draw a line through intercepts
Two-Point Form	y – 1 = 1/2 (x – 2)	– Plot points (2, 1) and (4, 2) – Use points to determine slope (1/2)

Finding the Equation of a Parabola from its Graph

A parabola is a U-shaped curve that opens either upward or downward. The equation of a parabola is typically written in the form y = ax² + bx + c, where a, b, and c are constants. To find the equation of a parabola from its graph, you need to know the coordinates of the vertex (the point where the parabola changes direction) and at least one other point on the curve.

Steps

Identify the vertex. The vertex is the highest or lowest point on the parabola. It is located at the point (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.
Find the slope of the line that passes through the vertex and another point on the curve. To do this, use the slope formula: m = (y₂ – y₁)/(x₂ – x₁), where (x₁, y₁) are the coordinates of the vertex and (x₂, y₂) are the coordinates of another point on the curve.
Substitute the vertex and slope into the equation y = mx + b. This will give you the equation of the line that passes through the vertex and another point on the curve.
Find the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis. To find the y-intercept, set x = 0 in the equation of the line.
Substitute the vertex and y-intercept into the equation y = ax² + bx + c. This will give you the equation of the parabola.

Example

Let’s say we have a parabola with a vertex at (2, -1) and another point on the curve at (3, 4). To find the equation of the parabola, we can follow the steps above:

Identify the vertex. The vertex is located at (2, -1).
Find the slope of the line that passes through the vertex and another point on the curve. Using the slope formula, we get:

m = (4 - (-1))/(3 - 2) = 5

Substitute the vertex and slope into the equation y = mx + b. We get:

y = 5x + b

Find the y-intercept of the line. Setting x = 0 in the equation of the line, we get:

y = 5(0) + b = b

Substitute the vertex and y-intercept into the equation y = ax² + bx + c. We get:

-1 = a(2)² + b(2) + c
-1 = 4a + 2b + c

We can also use the other point on the curve to get another equation:

4 = a(3)² + b(3) + c
4 = 9a + 3b + c

Solving these two equations simultaneously, we get:

a = -1
b = 3
c = 0

Therefore, the equation of the parabola is:

y = -x² + 3x

Finding the Coordinates of the Vertex from the Equation

The vertex of a parabola is the point where it changes direction. For a parabola in the form y = ax^2 + bx + c, the vertex is located at the point (-b/2a, -D/4a), where D is the discriminant, calculated as b^2 – 4ac.

In the case of y = 1/2x^2 – x, the vertex can be found as follows:

Step 1: Identify the Coefficients

a = 1/2

b = -1

Step 2: Calculate the Discriminant

D = b^2-4ac = (-1)^2-4(1/2)(0) = 1

Step 3: Find the x-Coordinate of the Vertex

x = -b/2a = -(-1)/2(1/2) = 1

Step 4: Find the y-Coordinate of the Vertex

y = -D/4a = -1/4(1/2) = -1/2

Step 5: Write the Coordinates of the Vertex

The vertex of the parabola y = 1/2x^2 – x is located at the point (1, -1/2).

Vertex Table

The coordinates of the vertex can be summarized in a table:

x-Coordinate	y-Coordinate
1	-1/2

Determining the Domain and Range of the Parabola

The domain of a function represents the set of all possible input values, while the range represents the set of all possible output values. In the context of the parabola given by the equation y = 1/2x, determining the domain and range involves analyzing the expression and considering its properties.

Domain

The domain of a function generally includes all real numbers unless there are specific restrictions imposed by the expression. In the case of y = 1/2x, there is no such restriction. The independent variable x can take on any real value, both positive and negative. Therefore, the domain of the parabola y = 1/2x is:

Domain: All real numbers (-∞, ∞)

Range

The range of a function depends on the shape and characteristics of the function’s graph. For the parabola y = 1/2x, understanding its range requires examining its behavior as x varies.

The parabola y = 1/2x is an open-down parabola. This means that as x increases, the values of y decrease. Conversely, as x decreases, the values of y increase.

However, there is one important limitation to the range of the parabola. As x approaches zero, the value of y approaches infinity. This is because the denominator of the expression becomes smaller and smaller as x gets closer to zero, causing the fraction to become larger and larger.

On the other hand, as x approaches negative or positive infinity, the value of y approaches zero. This is because the denominator of the expression becomes larger and larger as x gets further away from zero, causing the fraction to become smaller and smaller.

Therefore, the range of the parabola y = 1/2x is as follows:

Range: All real numbers except y = 0 (0, ∞) or (-∞, 0)

Tabulated Summary:

Property	Description
Domain	All real numbers (-∞, ∞)
Range	All real numbers except y = 0 (0, ∞) or (-∞, 0)

Sketching the Graph of Y = 1/2x Quickly

Graphing the equation y = 1/2x is a relatively simple process that can be done quickly and easily. By following these steps, you can create an accurate graph of this linear function in no time.

1. Find the Slope

The slope of a linear equation is a measure of its steepness. To find the slope of y = 1/2x, we can use the following formula:

“`
slope = m = -1/2
“`

The negative sign indicates that the graph of y = 1/2x will have a negative slope, meaning that it will slope downward from left to right.

2. Find the Y-Intercept

The y-intercept of a linear equation is the point where the graph crosses the y-axis. To find the y-intercept of y = 1/2x, we can substitute x = 0 into the equation:

“`
y = 1/2(0) = 0
“`

Therefore, the y-intercept of y = 1/2x is (0, 0).

3. Plot the Points

Now that we have the slope and the y-intercept, we can plot two points on the graph. One point will be on the y-axis, and the other point will be on the line itself.

The y-intercept is (0, 0), so we can plot that point first. To find a second point, we can use the slope to calculate the change in y for a given change in x. For example, if we move 2 units to the right (Δx = 2), the change in y will be:

“`
Δy = mΔx = (-1/2)(2) = -1
“`

Therefore, the second point is (2, -1).

4. Draw the Line

Once we have plotted two points on the graph, we can draw the line that passes through them. The line should have a negative slope, and it should pass through the y-intercept (0, 0).

5. Check Your Work

Once you have drawn the graph, it is always a good idea to check your work. You can do this by substituting a few different values of x into the equation and verifying that the resulting y-values match the points on your graph.

18. Applications of the Graph of Y = 1/2x

The graph of y = 1/2x has numerous applications in various fields, including physics, engineering, and economics. Here are a few examples:

a. Inverse Proportionality

The graph of y = 1/2x represents an inverse proportional relationship between y and x. This means that as x increases, y decreases, and vice versa. This relationship is often encountered in situations where two quantities are inversely proportional, such as the relationship between the distance traveled and the time taken to travel that distance.

b. Velocity and Time

In physics, the graph of y = 1/2x can be used to represent the relationship between velocity and time for an object moving with constant acceleration. The slope of the graph represents the acceleration, and the y-intercept represents the initial velocity.

c. Supply and Demand

In economics, the graph of y = 1/2x can be used to represent the relationship between supply and demand. The slope of the graph represents the elasticity of supply, and the y-intercept represents the equilibrium price.

d. Cost and Revenue

In business, the graph of y = 1/2x can be used to represent the relationship between cost and revenue. The slope of the graph represents the marginal cost, and the y-intercept represents the fixed cost.

e. Motion of a Projectile

In physics, the graph of y = 1/2x can be used to represent the trajectory of a projectile launched at a certain angle. The slope of the graph represents the range of the projectile, and the y-intercept represents the maximum height reached by the projectile.

Application	Description
Inverse Proportionality	Represents an inverse proportional relationship between two quantities.
Velocity and Time	Represents the relationship between velocity and time for an object moving with constant acceleration.
Supply and Demand	Represents the relationship between supply and demand in economics.
Cost and Revenue	Represents the relationship between cost and revenue in business.
Motion of a Projectile	Represents the trajectory of a projectile launched at a certain angle.

Graphing Y = 1/2x Using a Table of Values

To graph the linear equation y = 1/2x using a table of values, follow these steps:

Choose a set of x-values. Start with a few simple values of x, such as -2, -1, 0, 1, and 2.
Calculate the corresponding y-values. For each x-value, plug it into the equation y = 1/2x to find the corresponding y-value. For example, when x = -2, y = 1/2(-2) = -1.
Create a table of values. Organize the x- and y-values in a table, as shown below:

x	y
-2	-1
-1	-0.5
0	0
1	0.5
2	1

Plot the points. Mark each point on the coordinate plane using a small circle or dot.
Draw a line. Connect the points with a straight line. This line represents the graph of the equation y = 1/2x.

Section 20: Detailed Explanation of the Value 20

In the table of values, we chose the value 20 for x to illustrate how to find the corresponding y-value.

Plug 20 into the equation y = 1/2x.
y = 1/2(20)
y = 10
Confirm the result.
We can check our answer by substituting x = 20 and y = 10 back into the equation:
10 = 1/2(20)
10 = 10

Since the equation holds true, we know that the point (20, 10) lies on the graph of y = 1/2x.

Plot the point.
Mark the point (20, 10) on the coordinate plane, and connect it with a line to the other points to complete the graph.

Utilizing Intercepts to Plot the Graph Accurately

Intercepts are critical points where the graph of a function crosses the coordinate axes. Identifying the intercepts of the line y = 1/2x allows us to pinpoint two distinct points on the graph, providing a foundation for accurate plotting.

To find the x-intercept, we set y = 0 in the equation and solve for x:

Step	Equation
1	0 = 1/2x
2	x = 0

Therefore, the x-intercept is (0, 0).

To find the y-intercept, we set x = 0 in the equation and solve for y:

Step	Equation
1	y = 1/2(0)
2	y = 0

Thus, the y-intercept is (0, 0).

With both intercepts identified, we can now plot the graph of y = 1/2x:

Plot the x-intercept (0, 0) on the coordinate plane.
Plot the y-intercept (0, 0) on the coordinate plane.
Draw a straight line connecting the two intercepts. Since the slope of the line is 1/2, the line will rise 1 unit for every 2 units it moves to the right.

By utilizing the intercepts, we have effectively plotted the graph of y = 1/2x, ensuring its accuracy and providing a clear visual representation of the function.

Understanding the Asymptotes of the Parabola

Asymptotes are crucial lines that provide valuable information about the behavior of a parabola. They can help determine the shape and limits of the parabola, allowing us to better understand its overall characteristics. In the case of the parabola defined by the equation y = 1/2x, we have two types of asymptotes: vertical and horizontal.

Vertical Asymptote

The vertical asymptote is a vertical line that the parabola approaches but never intersects. For the equation y = 1/2x, the vertical asymptote is at x = 0. This is because as x approaches 0, the value of y approaches infinity or negative infinity, depending on whether x is positive or negative. The parabola will get closer and closer to the vertical asymptote, but it will never actually touch it.

Horizontal Asymptote

The horizontal asymptote is a horizontal line that the parabola approaches as x approaches positive or negative infinity. For the equation y = 1/2x, the horizontal asymptote is at y = 0. This is because as x becomes very large, either positively or negatively, the value of y approaches 0. The parabola will get closer and closer to the horizontal asymptote, but it will never actually intersect it.

Significance of Asymptotes

Asymptotes have several important implications for the graph of y = 1/2x:

* They divide the coordinate plane into regions where the parabola behaves differently.
* They help determine the domain and range of the parabola.
* They can be used to sketch the approximate shape of the parabola without plotting every point.
* They provide insights into the limits and behavior of the function as x approaches certain values.

Understanding the asymptotes of a parabola is essential for fully comprehending its graph and behavior. By identifying the vertical and horizontal asymptotes for the equation y = 1/2x, we gain valuable information about the shape, limits, and tendencies of this particular parabola.

Interpreting the Slope of the Parabola

The slope of a parabola is a crucial element in understanding the parabola’s shape, direction, and rate of change. The slope of a parabola is represented by the numerical coefficient (a) that accompanies the x-term in the quadratic equation. In this case, for the parabola y = 1/2x^2, the slope is 24, which plays a significant role in determining the parabola’s characteristics.

Impact of Slope on Parabola Shape

The slope of a parabola primarily affects its concavity. A positive slope, like in this case, indicates that the parabola opens upward, resembling a U-shaped curve. Conversely, a negative slope would result in a parabola opening downward, forming an inverted U-shape.

Determining the Direction of Opening

The slope of a parabola also provides valuable information about its direction of opening. A positive slope signifies that the parabola opens upward, with the vertex pointing toward the positive y-axis. In contrast, a negative slope indicates that the parabola opens downward, with the vertex pointing toward the negative y-axis.

Measuring the Steepness

The magnitude of the slope governs the steepness of the parabola’s curvature. A larger slope, such as 24 in this instance, indicates that the parabola is more sharply curved, resulting in a narrower, more compact shape. A smaller slope would yield a parabola with a gentler curvature and a wider, more elongated shape.

Calculating the Rate of Change

The slope of a parabola is intimately connected to the rate of change of the parabola. The slope represents the vertical change in the y-coordinate relative to the horizontal change in the x-coordinate. In this case, for the parabola y = 1/2x^2, the slope of 24 implies that when x increases by 1 unit, the corresponding y-coordinate increases by 24 units. This value represents the rate of change or the gradient of the parabola.

Example

Consider the parabola y = 1/2x^2, which has a slope of 24. This means that the parabola opens upward and is more sharply curved than a parabola with a smaller slope. The rate of change for this parabola is 24, indicating that for each unit increase in x, the y-coordinate increases by 24 units.

Significance of the Slope

The slope of a parabola is a fundamental characteristic that influences the parabola’s overall appearance, behavior, and rate of change. Understanding the slope enables you to interpret the parabola’s shape, direction of opening, curvature, and rate of change accurately.

Slope	Opening	Rate of Change
Positive	Upward	Positive
Negative	Downward	Negative
Larger Positive	Sharply Curved	Steeper
Smaller Positive	Gently Curved	Less Steep
Larger Negative	Sharply Curved Downward	Steeper Downward
Smaller Negative	Gently Curved Downward	Less Steep Downward

Graphing Y = 1/2x by Completing the Square

To graph the equation y = 1/2x using the method of completing the square, follow these steps:

Rewrite the equation: Multiply both sides of the equation by 2 to get rid of the fraction:

2y = 1x

Complete the square: To complete the square, add and subtract the square of half the coefficient of x, which is (1/2)^2 = 1/4:

2y + 1/4 – 1/4 = 1x

Factor the perfect square trinomial: The left-hand side of the equation can be factored as a perfect square trinomial:

2y + 1/4 = (√2y)^2 – (1/2)^2

Write the equation in vertex form: The vertex form of a parabola is:

y = a(x – h)^2 + k

where (h, k) is the vertex of the parabola.

Substituting the values from the previous step, we get:

2y = (√2y – 1/2)^2

Dividing both sides by 2, we get:

y = (1/2)(√2y – 1/2)^2

Identify the vertex: The vertex of the parabola is (1/2, 0).
Plot the vertex: Plot the point (1/2, 0) on the coordinate plane.
Find additional points: To find additional points on the parabola, you can use the equation:

y = (1/2)(√2y – 1/2)^2

For example, you can find the x-intercepts by setting y = 0 and solving for x:

0 = (1/2)(√2(0) – 1/2)^2

0 = (-1/4)^2

x = ±1/4

So the x-intercepts are (-1/4, 0) and (1/4, 0).

Sketch the parabola: Use the vertex, the x-intercepts, and any other points you have found to sketch the parabola.

Here is a table summarizing the steps required to graph y = 1/2x using the method of completing the square:

Step	Description
1	Multiply both sides of the equation by 2 to get rid of the fraction
2	Add and subtract the square of half the coefficient of x
3	Factor the perfect square trinomial
4	Write the equation in vertex form
5	Identify the vertex
6	Plot the vertex
7	Find additional points
8	Sketch the parabola

Determining the Symmetry Point of the Parabola

The symmetry point of a parabola, also known as the vertex, is the point where the parabola changes direction. It is the lowest point on a parabola that opens upward or the highest point on a parabola that opens downward. The symmetry point is a key feature of a parabola and can be used to determine other important characteristics of the graph.

Finding the Symmetry Point

To find the symmetry point of a parabola, you need to first determine the equation of the parabola. The equation of a parabola is typically written in the form y = ax^2 + bx + c, where a, b, and c are constants. Once you have the equation of the parabola, you can use the following steps to find the symmetry point:

1. Set the derivative of the parabola equal to zero. The derivative of a parabola is equal to 2ax + b. To find the symmetry point, you need to set the derivative equal to zero and solve for x.
2. Solve for x. Once you have set the derivative equal to zero, you can solve for x by dividing both sides of the equation by 2a.
3. Plug x back into the equation of the parabola. Once you have solved for x, you can plug the value of x back into the equation of the parabola to find the y-coordinate of the symmetry point.

Example

Let’s find the symmetry point of the parabola y = x^2 – 4x + 3.

1. Set the derivative of the parabola equal to zero. The derivative of this parabola is y’ = 2x – 4. To set the derivative equal to zero, we can write: 2x – 4 = 0.

2. Solve for x. Solving for x, we get: x = 2.

3. Plug x back into the equation of the parabola. Plugging x = 2 back into the equation of the parabola, we get: y = 2^2 – 4(2) + 3 = -1.

Therefore, the symmetry point of the parabola y = x^2 – 4x + 3 is (2, -1).

Properties of the Symmetry Point

The symmetry point of a parabola has several important properties:

The symmetry point is the turning point of the parabola. This means that the parabola changes direction at the symmetry point.
The symmetry point is the midpoint of the line that connects the x-intercepts of the parabola.
The symmetry point is the vertex of the parabola. This means that the parabola is symmetric about the symmetry point.

Summary

The symmetry point of a parabola is a key feature of the graph. It is the point where the parabola changes direction and is the turning point of the parabola. The symmetry point can be found by setting the derivative of the parabola equal to zero and solving for x. The symmetry point has several important properties, including being the midpoint of the line that connects the x-intercepts of the parabola and being the vertex of the parabola.

Property	Description
Turning point	The point where the parabola changes direction.
Midpoint of x-intercepts	The point that is halfway between the two x-intercepts of the parabola.
Vertex	The highest or lowest point on the parabola.

Finding the Focus and Directrix of the Parabola

To determine the focus and directrix of the parabola represented by the equation y = 1/2x^2, we need to first identify the vertex (h, k) of the parabola. Since the equation is in the form y = a(x – h)^2 + k, we can directly read off the vertex as (0, 0).

Next, we need to determine the value of “a” from the equation, which is 1/2. The value of “a” determines the vertical stretch or compression of the parabola.

Focus

The focus of a parabola is a point (h + p, k) which is p units from the vertex along the axis of the parabola. Since the parabola opens upwards (i.e., the coefficient of x^2 is positive), the axis of the parabola is the y-axis. Therefore, the focus will be located at a distance of p units above the vertex.

To determine the value of “p,” we use the formula p = 1/4a. Substituting the value of “a” (1/2) into this formula, we get:

“`
p = 1/4 * 1/2 = 1/8
“`

Therefore, the focus of the parabola y = 1/2x^2 is located at the point (0, 1/8).

Directrix

The directrix of a parabola is a horizontal line located at a distance of p units below the vertex. In this case, the directrix will be located at a distance of 1/8 units below the vertex (0, 0).

The equation of the directrix is y = k – p, where (h, k) is the vertex. Substituting the values of h and k (both equal to 0) and p (1/8), we get:

“`
y = 0 – 1/8
“`

Therefore, the equation of the directrix is y = -1/8.

Table summarizing the focus and directrix:


Focus:	(0, 1/8)
Directrix:	y = -1/8

Graphing Y = 1/2x Using Transformations

Step 1: Graph Y = 1/x

The parent function for Y = 1/2x is Y = 1/x, which is a hyperbola. To graph it:

Plot the point (1, 1) on the coordinate plane.
From this point, move 1 unit to the right and 1 unit up to plot (2, 1/2).
Repeat this process until you have several points on the upper and lower branches of the hyperbola.
Connect the points with a smooth curve to complete the graph.

Step 2: Shrink the Graph Vertically

To transform Y = 1/x to Y = 1/2x, we need to shrink the graph vertically by a factor of 2.

For each point (x, y) on the Y = 1/x graph, the corresponding point on the Y = 1/2x graph will be (x, y/2).
For example, the point (2, 1/2) on the Y = 1/x graph becomes (2, 1/4) on the Y = 1/2x graph.
Plot the new points and connect them with a smooth curve to complete the graph of Y = 1/2x.

Additional Notes on Vertical Shrinkage

Vertical shrinkage does not affect the shape of the graph.
It only scales the height of the graph relative to the y-axis.
If the vertical shrinkage factor is greater than 1, the graph will become wider than the original.
If the vertical shrinkage factor is less than 1, the graph will become narrower than the original.

Transformation	Effect on Graph
Shrink vertically by a factor of 2	Graph becomes narrower by a factor of 2
Shrink vertically by a factor of 1/2	Graph becomes wider by a factor of 2

The general equation for vertical shrinkage is:
$$
y = \frac{1}{a} f(x)
$$
where:
$$a$$ is the vertical shrinkage factor.

Scaling the Graph of Y = 1/2x

31. The Effect of Multiplying the Independent Variable by a Constant: Transformations with Respect to the x-axis

When the independent variable (x) is multiplied by a constant (a), the graph of the function (y = f(ax)) undergoes the following transformations:

Scaling in the x-direction: The graph is stretched or compressed horizontally by a factor of (1/a).

Reflection over the x-axis: If (a) is negative, the graph is reflected over the x-axis.

To illustrate this, let’s consider the function (y = 1/2x) as an example.

a) Scaling in the x-direction:

If (a > 1) (e.g., (a = 2)), the graph of (y = 1/2(2x)) is stretched horizontally by a factor of (1/2). This means that the x-coordinates of the points on the graph are halved.
If (0 < a < 1) (e.g., (a = 0.5)), the graph of (y = 1/2(0.5x)) is compressed horizontally by a factor of (1/0.5 = 2). This means that the x-coordinates of the points on the graph are doubled.

b) Reflection over the x-axis:

If (a) is negative (e.g., (a = -2)), the graph of (y = 1/2(-2x)) is reflected over the x-axis. This means that the points on the graph are mirrored across the x-axis.

To summarize these transformations, the following table shows the effects of multiplying the independent variable (x) by different constants:

Constant $a$, (value of x-scaling factor, 1/a)	Transformation
$a > 1$	Stretched horizontally by a factor of $1/a$
$0 < a < 1$	Compressed horizontally by a factor of $1/a$
$a < 0$	Reflected over the x-axis and stretched horizontally by a factor of $1/a$

Stretching or Compressing the Graph of Y = 1/2x

The graph of Y = 1/2x is a hyperbola that opens to the left and right. We can stretch or compress this graph by multiplying the x-coordinate of each point by a constant. If the constant is greater than 1, the graph will be stretched. If the constant is between 0 and 1, the graph will be compressed.

For example, let’s graph the function Y = 1/2x.

x	y
-2	-1
-1	-2
0	0
1	2
2	1

Now let’s graph the function Y = 1/x.

x	y
-2	-1/2
-1	-1
0	undefined
1	1
2	1/2

As you can see, the graph of Y = 1/x is stretched in the x-direction compared to the graph of Y = 1/2x. This is because the constant 1 is greater than 1.

We can also compress the graph of Y = 1/2x by multiplying the x-coordinate of each point by a constant between 0 and 1. For example, let’s graph the function Y = 1/4x.

x	y
-2	-1/4
-1	-1/2
0	0
1	1/4
2	1/2

As you can see, the graph of Y = 1/4x is compressed in the x-direction compared to the graph of Y = 1/2x. This is because the constant 1/4 is between 0 and 1.

Rotating the Graph of Y = 1/2x

To rotate the graph of Y = 1/2x, we need to apply a transformation to the equation. The transformation will involve rotating the graph around the origin by a certain angle. The angle of rotation is typically measured in degrees or radians.

The formula for rotating a point (x, y) around the origin by an angle θ is as follows:

“`
x’ = x cos(θ) – y sin(θ)
y’ = x sin(θ) + y cos(θ)
“`

Where (x’, y’) are the coordinates of the rotated point.

To rotate the graph of Y = 1/2x by an angle of 45 degrees, we can apply the following transformation:

“`
x’ = x cos(45°) – y sin(45°)
y’ = x sin(45°) + y cos(45°)
“`

Substituting the equation of the line into the transformation equations, we get the following:

“`
x’ = x cos(45°) – (1/2x) sin(45°)
y’ = x sin(45°) + (1/2x) cos(45°)
“`

Simplifying the equations, we get the following:

“`
x’ = (√2/2) x – (√2/4) y
y’ = (√2/4) x + (√2/2) y
“`

This is the equation of the rotated graph. The graph has been rotated by 45 degrees around the origin.

Steps to Rotate the Graph of Y = 1/2x by 45 Degrees

Substitute the equation of the line into the transformation equations.
Simplify the equations to get the equation of the rotated graph.
Graph the rotated graph.

Example

To graph the graph of Y = 1/2x rotated by 45 degrees around the origin, follow these steps:

Substitute the equation of the line into the transformation equations.

x’ = x cos(45°) – (1/2x) sin(45°)

y’ = x sin(45°) + (1/2x) cos(45°)

Simplify the equations to get the equation of the rotated graph.

x’ = (√2/2) x – (√2/4) y

y’ = (√2/4) x + (√2/2) y

Graph the rotated graph.

The graph of the rotated graph is shown below.

Identifying the Vertex of the Inverse Function

Converting the Equation of the Hyperbola to the Equation of the Inverse Function

To identify the vertex of the inverse function, we first need to convert the equation of the hyperbola to the equation of the inverse function. The general equation of a hyperbola is:

$$y^2 – {b^2}{x^2} = {a^2}$$

Given the equation $y = 1 + 2x$, we can identify $a$ and $b$ as follows:

$$a = 1, b = 2$$

The equation of the inverse function is:

$$x = 1 + 2y$$

Finding the Vertex of the Inverse Function

The vertex of a hyperbola in the form $x^2 – {b^2}{y^2} = {a^2}$ is given by the point $(0, \pm a)$.

In our case, the inverse function is in the form $y^2 – {2^2}{x^2} = {1^2}$, so the vertex of the inverse function is:

$$(0, \pm 1)$$

Therefore, the vertex of the inverse function $x = 1 + 2y$ is at the point (0, 1).

Finding the Intercepts of the Inverse Function

Finding the x-intercept of f^-1(x)

To find the x-intercept of the inverse function f^-1(x), we set y = 0 and solve for x:

“`
0 = 1/2x
x = 0
“`

Therefore, the x-intercept is (0, 0).

Finding the y-intercept of f^-1(x)

To find the y-intercept of the inverse function f^-1(x), we set x = 0 and solve for y:

“`
y = 1/2(0)
y = 0
“`

Therefore, the y-intercept is also (0, 0).

Summary of the intercepts of f^-1(x)

The intercepts of the inverse function f^-1(x) are summarized in the following table:

Intercept	Coordinates
x-intercept	(0, 0)
y-intercept	(0, 0)

Graphical interpretation of the intercepts of f^-1(x)

The intercepts of the inverse function f^-1(x) provide important information about its graph:

* The x-intercept represents the point where the graph of f^-1(x) intersects the x-axis.
* The y-intercept represents the point where the graph of f^-1(x) intersects the y-axis.
* The fact that both intercepts are at the origin indicates that the graph of f^-1(x) is symmetric with respect to the line y = x.

Plotting Points on the Inverse Function

To graph the inverse function, we need to find the inverse of the original function, which is $y = 1/2x$. To do this, we swap the roles of $x$ and $y$ and solve for $y$.

$$x = 1/2y$$
$$2x = y$$
$$y = 2x$$

So the inverse function is $y = 2x$. Now we can plot points on the inverse function just like we did with the original function. Let’s start by finding the $x$- and $y$-intercepts.

1. Find the $x$-intercept:
Set $y = 0$ and solve for $x$.

$$0 = 2x$$
$$x = 0$$

So the $x$-intercept is $(0, 0)$.

2. Find the $y$-intercept:
Set $x = 0$ and solve for $y$.

$$y = 2(0)$$
$$y = 0$$

So the $y$-intercept is $(0, 0)$.

3. Plot the points:
We can now plot the points $(0, 0)$ and any other points that we want to find. For example, we could plot the point $(1, 2)$ by substituting $x = 1$ into the equation $y = 2x$.

$$y = 2(1)$$
$$y = 2$$

So the point $(1, 2)$ is on the graph of the inverse function.

4. Draw the line:
Once we have plotted a few points, we can draw the line that passes through them. The line should be a straight line with a slope of 2.

The graph of the inverse function is shown below.

Summary of steps to plot points on the inverse function:

Find the inverse function.
Find the $x$- and $y$-intercepts.
Plot the intercepts and any other points that you want to find.
Draw the line that passes through the points.

Determining the Slope and y-Intercept of Y = 2x

To graph any linear equation, we need to determine its slope and y-intercept. For the equation Y = 2x, the slope is 2 and the y-intercept is 0. This information will help us plot the graph accurately.

Slope: 2

The slope represents the rate of change in the y-coordinate relative to the change in the x-coordinate. In other words, it tells us how much the y-coordinate increases (or decreases) for every unit increase (or decrease) in the x-coordinate. In our case, the slope is 2, which means that for every unit increase in x, the y-coordinate increases by 2 units.

y-Intercept: 0

The y-intercept is the point where the graph crosses the y-axis. It represents the value of y when x is equal to 0. In our equation, the y-intercept is 0, which means that the graph intersects the y-axis at the point (0, 0).

Plotting the Graph

Now that we know the slope and y-intercept, we can start plotting the graph. We will use a coordinate plane with x and y axes.

Step 1: Plot the y-Intercept

Start by plotting the y-intercept, which is (0, 0). This point represents the point where the graph crosses the y-axis.

Step 2: Determine the Slope

The slope of the equation is 2, which means that for every unit increase in x, the y-coordinate increases by 2 units.

Step 3: Use the Slope to Find Additional Points

From the y-intercept (0, 0), we can use the slope to determine additional points on the graph.

Increase x by 1 unit and increase y by 2 units. This gives us the point (1, 2).
Increase x by another 1 unit and increase y by 2 units again. This gives us the point (2, 4).

Step 4: Draw the Line

Plot the additional points (1, 2) and (2, 4) on the coordinate plane. Connect these points with a straight line. This line represents the graph of Y = 2x.

Step 5: Check for Accuracy

Make sure the line passes through the y-intercept (0, 0) and has the correct slope of 2. Also, check that the line passes through the additional points (1, 2) and (2, 4). This verification ensures the accuracy of the graph.

Shifting the Graph of the Inverse Function Horizontally

Step 41: Shifting the Graph of f-1(x) to the Left by h Units

To shift the graph of f-1(x) h units to the left, we need to replace x in the equation of f-1(x) with (x + h). This is because shifting the graph to the left means that for any given value of y, the corresponding value of x will be h units less than it would be on the original graph.

For example, if we want to shift the graph of f-1(x) = 2x + 1 to the left by 3 units, we would replace x with (x + 3) in the equation, giving us the equation f-1(x) = 2(x + 3) + 1 = 2x + 7.

The following table summarizes the steps involved in shifting the graph of f-1(x) h units to the left:

Step	Equation
Replace x with (x + h) in the equation of f-1(x).	f-1(x + h) = 2x + 1
Simplify the equation.	f-1(x + h) = 2x + 7

Step 42: Shifting the Graph of f-1(x) to the Right by h Units

To shift the graph of f-1(x) h units to the right, we need to replace x in the equation of f-1(x) with (x – h). This is because shifting the graph to the right means that for any given value of y, the corresponding value of x will be h units greater than it would be on the original graph.

For example, if we want to shift the graph of f-1(x) = 2x + 1 to the right by 3 units, we would replace x with (x – 3) in the equation, giving us the equation f-1(x) = 2(x – 3) + 1 = 2x – 5.

The following table summarizes the steps involved in shifting the graph of f-1(x) h units to the right:

Step	Equation
Replace x with (x – h) in the equation of f-1(x).	f-1(x – h) = 2x + 1
Simplify the equation.	f-1(x – h) = 2x – 5

Reflecting the Graph of the Inverse Function over the y-axis

When reflecting a graph over the y-axis, we essentially flip the graph around the vertical axis. This operation essentially replaces x with -x, as every point’s distance from the y-axis changes its sign.

To reflect the graph of the inverse function over the y-axis, we follow these steps:

Replace x with -x in the function’s equation:

y = 1 / (2x) becomes y = 1 / (2(-x)) = 1 / (-2x)

Simplify the equation:

y = 1 / (-2x) = -1 / (2x)

The resulting equation, y = -1 / (2x), represents the inverse function reflected over the y-axis.

Analyzing the Reflected Graph

The reflected graph of the inverse function is a vertical stretch of the original inverse function by a factor of 1. This means that the graph is stretched away from the x-axis by a factor of 1.

The reflected graph also has a reflection symmetry about the y-axis. This means that for any point (x, y) on the graph, the point (-x, y) is also on the graph.

Key Points

Reflecting a graph over the y-axis flips the graph around the vertical axis.
To reflect the graph of an inverse function over the y-axis, replace x with -x in the function’s equation.
The reflected graph is a vertical stretch of the original inverse function by a factor of 1 and has a reflection symmetry about the y-axis.

Original Function	Inverse Function	Reflected Inverse Function
y = 2x	y = 1 / (2x)	y = -1 / (2x)

Combining Transformations for the Inverse Function

So far, we have only worked with individual transformations of the function $y = f(x)$. In this section, we will explore what happens when we combine two or more transformations. For example, we may have a function that is vertically stretched and then shifted to the left. To graph this function, we would perform the transformations in the order in which they are given. First, we would stretch the function vertically, and then we would shift it to the left.

Combining transformations can be a bit tricky, but with practice, it becomes easier. The key is to remember that the transformations are applied in the order in which they are written. For example, the function $y = 2f(x – 3)$ is first shifted 3 units to the right and then vertically stretched by a factor of 2.

Horizontal vs. Vertical Transformations

When combining transformations, it is important to distinguish between horizontal and vertical transformations. Horizontal transformations affect the x-coordinates of the graph, while vertical transformations affect the y-coordinates of the graph. For example, the function $y = f(x + 3)$ is a horizontal translation, while the function $y = 2f(x)$ is a vertical stretch.

When combining horizontal and vertical transformations, the order in which the transformations are applied matters. For example, the function $y = 2f(x – 3)$ is first shifted 3 units to the right and then vertically stretched by a factor of 2. However, the function $y = f(x – 3) + 2$ is first shifted 3 units to the right and then vertically translated 2 units up.

Reflections

Reflections are another type of transformation that can be applied to functions. A reflection over the x-axis changes the sign of the y-coordinate of every point on the graph, while a reflection over the y-axis changes the sign of the x-coordinate of every point on the graph.

Reflections can be combined with other transformations to create even more complex graphs. For example, the function $y = -f(x – 3)$ is a reflection over the x-axis and a horizontal translation 3 units to the right.

Example

Graph the function $y = |x – 2| + 1$.

First, we will graph the function $y = |x|$. This function is a V-shaped graph with vertex at the origin. Next, we will translate the graph 2 units to the right. This will give us the graph of the function $y = |x – 2|$. Finally, we will translate the graph 1 unit up. This will give us the graph of the function $y = |x – 2| + 1$.

The following table summarizes the transformations that were applied to the function $y = |x|$ to create the graph of the function $y = |x – 2| + 1$.

Transformation	Effect on the Graph
Translate 2 units to the right	Shifts the graph 2 units to the right
Translate 1 unit up	Shifts the graph 1 unit up

Applications of the Graph of Y = 2x

The graph of the function y = 2x is a straight line that passes through the origin and has a slope of 2. This graph can be used to solve a variety of problems, including:

• Finding the equation of a line.

• Graphing a linear equation.

• Solving systems of linear equations.

• Finding the slope of a line.

• Determining the y-intercept of a line.

• Calculating the area of triangles and parallelograms.

• Solving word problems involving linear equations.

46. Finding the x-intercept of a line

The x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept of the line y = 2x, set y = 0 and solve for x:

0 = 2x
x = 0

Therefore, the x-intercept of the line y = 2x is (0, 0).

Here is a table summarizing the key points about the graph of y = 2x:

Point	y-value
(0, 0)	0
(1, 2)	2
(2, 4)	4
(-1, -2)	-2
(-2, -4)	-4

The graph of y = 2x is a straight line that passes through the origin and has a slope of 2. This graph can be used to solve a variety of problems, including those listed above.

Finding the Coordinates of the Vertex of the Inverse Function from the Equation

Step 1: Solve the equation for x in terms of y

Rewrite the given equation, y = 1/2x, as:

x = 2y

Step 2: Swap x and y

To find the inverse function, swap the roles of x and y:

y = 2x

Step 3: Find the equation of the vertex

The vertex of a parabola is always at the point (h, k) where h is the x-coordinate and k is the y-coordinate. For the equation y = 2x, the vertex is at the point:

(0, 0)

Step 4: Determine the shape of the inverse function

Since the coefficient of x in the equation y = 2x is positive, the parabola opens upward. This means that the inverse function will be a parabola that opens downward.

Step 5: Determine the direction of the opening

Since the coefficient of x in the equation y = 2x is positive, the parabola opens upward. This means that the inverse function will open downward.

Step 6: Determine the coordinates of the vertex of the inverse function

Since the inverse function is a parabola that opens downward with a vertex at (0, 0), the inverse function has a vertex at:

(0, 0)

Summary Table:

Original Function	Inverse Function
y = 1/2x	y = 2x
Vertex: (0, 0)	Vertex: (0, 0)

Determining the Domain and Range of the Inverse Function

The inverse of a function is a new function that "undoes" the original function. To determine the domain and range of the inverse function, we need to switch the roles of the input (x) and output (y) variables.

Step 1: Solve for y

Begin by isolating the output variable (y) in the original function:

y = 1/2x

2y = x

Step 2: Swap x and y

Interchange the roles of x and y to obtain the inverse function:

x = 2y

Step 3: Determine the Domain of the Inverse Function

The domain of the inverse function comprises all possible values of x in the original function. From the original function, we observe that x can take on any real number except zero:

x ≠ 0

Therefore, the domain of the inverse function is:

x ∈ (-∞, 0) ∪ (0, ∞)

Step 4: Determine the Range of the Inverse Function

The range of the inverse function comprises all possible values of y in the original function. From the original function, we observe that y can take on any real number:

y ∈ (-∞, ∞)

Therefore, the range of the inverse function is:

y ∈ (-∞, ∞)

Step 5: Properties of the Inverse Function

The inverse function shares several properties with the original function:

Symmetry about the line y = x: The graph of the inverse function is a reflection of the original function over the line y = x.
Linear Function: Both the original function and its inverse are linear functions with a slope of 1/2.
Reciprocal Relationship: The inverse function is the reciprocal of the original function:

f(x) = 1/2x
g(x) = 2x

where g(x) is the inverse function of f(x).

How to Graph y = 1/2x

To graph the equation y = 1/2x, follow these steps:

Plot the intercepts. The y-intercept is found by setting x = 0. y = 1/2(0) = 0, so the y-intercept is (0, 0). The x-intercept is found by setting y = 0. 0 = 1/2x, so x = 0. Therefore, the x-intercept is (0, 0).
Draw a line through the intercepts. The line passes through the points (0, 0).
Check your graph. You can check your graph by plugging in a few points to make sure they satisfy the equation. For example, when x = 1, y = 1/2(1) = 1/2. So the point (1, 1/2) is on the line.

Constant \(a\), (value of x-scaling factor, 1/a)	Transformation
\(a > 1\)	Stretched horizontally by a factor of \(1/a\)
\(0 < a < 1\)	Compressed horizontally by a factor of \(1/a\)
\(a < 0\)	Reflected over the x-axis and stretched horizontally by a factor of \(1/a\)