Modify Expressions to Translate Graphs

7.17.2 Modify Expressions to Translate Graphs

7.17.2 • Modify Expressions to Translate Graphs

Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

1. How would you change the equation $f (x) = x^{2}$ so that the vertex of the graph of the new equation is located at the following coordinates and the graph opens as described?

a. $(0, 11)$ , opens up

Compare your answer:

$f (x) = x^{2} + 11$

b. $(7, 11)$ , opens up

Compare your answer:

Answer: $f (x) = (x - 7)^{2} + 11$

c. $(7, - 3)$ , opens down

Compare your answer:

Answer: $f (x) = - 1 (x - 7)^{2} - 3$

2. Use the graphing tool or technology outside the course. Verify your predictions.

Access multimedia content

Compare your answer:

A graph with four labeled parabolas with vertices at (0, 11), (0, 0), (7, negative 3), and at (7, 11).

3. Kiran graphed the equation $f (x) = x^{2} + 1$ and noticed that the vertex is at $(0, 1)$ . He changed the equation to $f (x) = (x - 3)^{2} + 1$ and saw that the graph shifted 3 units to the right and the vertex is now at $(3, 1)$ . Next, he graphed the equation $f (x) = x^{2} + 2 x + 1$ and observed that the vertex is at $(- 1, 0)$ . Kiran thought, "If I change the squared term $x^{2}$ to $(x - 5)^{2}$ , the graph will move 5 units to the right and the vertex will be at $(4, 0)$ ."

a. Do you agree with Kiran?

Disagree

b. Explain or show your reasoning why you agree or disagree with Kiran.

Compare your answer:

You should disagree with Kiran.

Four parabolas on a graph with labeled vertices: black at (negative 1,0), orange at (0,1), purple at (3,1), and red at (4,10).

The vertex of the graph of the original equation $f (x) = x^{2} + 2 x + 1$ is at $(- 1, 0)$ . The vertex of $f (x) = (x - 5)^{2} + 2 x + 1$ is at $(4, 10)$ .
Evaluating $f (x) = (x - 5)^{2} + 2 x + 1$ at $x = 4$ gives $f (4) = 10$ , not $f (4) = 0$ .
The original equation is not in vertex form, so changing $x^{2}$ into $(x - 5)^{2}$ does not change the location of the vertex 5 units to the right.

4. $f (x) = (x + 4)^{2} - 1$ is a parabola with a vertex at $(- 4, 1)$ that opens upward.

a. If we multiply $f (x)$ by 2, we get $2 f (x) = 2 [(x + 4)^{2} - 1] = 2 (x + 4)^{2} - 2$ . How does this change the graph?

Compare your answer:

The parabola is narrower and the vertex is lower. The $x$ -intercepts stay the same.

b. If we multiply $f (x)$ by $\frac{1}{2}$ , we get $\frac{1}{2} f (x) = \frac{1}{2} [(x + 4)^{2} - 1] = \frac{1}{2} (x + 4)^{2} - \frac{1}{2} .$ How does this change the graph?

Compare your answer:

The parabola is wider and the vertex is higher. The $x$ -intercepts stay the same.

5. Use the graphing tool or technology outside the course. Verify your prediction using Desmos tool below.

Access multimedia content

Compare your answer:

Graphs of three parabolas that illustrate how a parabola has been changed after applying a vertical stretch factor of 2 and a vertical compression factor of 2 to the function's equation.

6. Sophia graphed the equation $f (x) = x^{2} - 6 x + 9$ . She wondered what would happen to the graph if everywhere she had an $x$ , she replaced it with $3 x$ .

a. Sophia predicted that the parabola would be narrower, and the $x$ -intercepts and $y$ -intercepts would change. Write why you agree or disagree with Sophia’s prediction.

Compare your answer:

The parabola will be narrower and the $x$ -intercepts will change, but the $y$ -intercept will stay the same.

b. Sophia also wondered what would happen if she replaced the $x$ in $f (x) = x^{2} - 6 x + 9$ with $\frac{1}{3} x$ , how that would change the graph. She predicted that the parabola would be wider, and the $x$ -intercepts and $y$ -intercepts would change. Write why you agree or disagree with Sophia’s prediction.

Compare your answer:

The parabola will be wider and the $x$ -intercepts will change, but the $y$ -intercept will stay the same.

7. Use the graphing tool or technology outside the course. Verify your prediction using Desmos tool below.

Access multimedia content

Compare your answer:

Graphs of three parabolas that illustrate how a parabola has been changed after applying a horizontal stretch factor of 3 and a horizontal compression factor of 3 to the function's equation.

Self Check

What transformations and shifts need to be made to change

y=x^2

y=-(x+5)^2-10

Reflect over the $x$ -axis, left 5, and up 10
Reflect over the $x$ -axis, left 5, and down 10
Left 5 and down 10
Right 5 and down 10

Additional Resources

Transforming Graphs of Quadratics

The graphs of $f (x) = x^{2}$ , $f (x) = x^{2} + 12$ , and $f (x) = (x + 3)^{2}$ all have the same shape but their locations are different. The graph that represents $f (x) = x^{2}$ has its vertex at $(0, 0)$ .

Graph of the parent quadratic function is given in blue on the coordinate plane. A function that has been shifted horizontally is graphed in red. A function that has been shifted vertically is graphed in green.

Notice that adding 12 to $x^{2}$ raises the graph by 12 units, so the vertex of that graph is at $(0, 12)$ . Replacing $x^{2}$ with $(x + 3)^{2}$ shifts the graph 3 units to the left, so the vertex is now at $(- 3, 0)$ .

We can also shift a graph both horizontally and vertically.

The graph that represents $f (x) = (x + 3)^{2} + 12$ will look like that for $f (x) = x^{2}$ but it will be shifted 12 units up and 3 units to the left. Its vertex is at $(- 3, 12)$ .

Graph of the parent quadratic function is given in blue on the coordinate plane. A function that has been shifted both horizontally and vertically is graphed in gold.

Translating Functions

A vertical shift transforms the parent function into another function by moving the graph up or down “d” units.

A horizontal shift transforms the parent function into another function by moving the graph left or right “c” units.

The graph representing the equation $f (x) = - (x + 3)^{2} + 12$ has the same vertex at $(- 3, 12)$ , but because the squared term $(x + 3)^{2}$ is multiplied by a negative number, the graph is flipped over horizontally, so that it opens downward.

A function and its reflection are graphed on a coordinate plane. The gold graph is labeled y equals the quantity of x plus 3 squared plus 12. The black graph is labeled y equals the opposite of the quanitity of x plus 3 squared plus 12.

Quadratic equations can also be transformed through dilations that stretch or compress the parabola. The graphs for the parent quadratic function, $f (x) = x^{2}$ , and $f (x) = 4 x^{2}$ , and $f (x) = \frac{1}{5} x^{2}$ are all parabolas. The second and third equation, however, have been vertically dilated from the parent function.

Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been vertically dilated are graphed as the orange and green parabolas.

The orange dashed parabola represents $f (x) = 4 x^{2}$ and depicts the result of multiplying each output of the function by 4. This dilation represents a vertical stretch.

The green dotted parabola represents $f (x) = \frac{1}{5} x^{2}$ and depicts a graph where each of the output values from the function are multiplied by $\frac{1}{5}$ . This dilation represents a vertical compression.

The scale factor of the coefficient we multiply by determines the impact on the graph. If the scale factor is greater than 1, then the dilation is a vertical stretch. If the scale factor is between 0 and 1, then the dilation is a vertical compression.

When quadratic equations are dilated horizontally, the result looks similar but the impact to the function is very different because horizontal dilations affect the input values before the function is applied. The graphs for the parent quadratic function, $f (x) = x^{2}$ , and $f (x) = (4 x)^{2}$ and $f (x) = (\frac{1}{3} x)^{2}$ are displayed below.

Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been horizontally dilated are graphed as the orange and green parabolas.

The orange dashed parabola represents $f (x) = (4 x)^{2}$ and depicts the result of multiplying each input of the function by 4 and then squaring. This dilation represents a horizontal compression.

The green dotted parabola represents $f (x) = (\frac{1}{3} x)^{2}$ and depicts a graph where each of the input values from the function are multiplied by $\frac{1}{3}$ . This dilation represents a horizontal stretch.

Notice that the scale factor determines the impact on the graph here, too. If the scale factor is greater than 1, then the dilation is a horizontal compression. If the scale factor is between 0 and 1, then the dilation is a horizontal stretch.

Dilating Functions

$a f (b x)$

A vertical dilation transforms the parent function into another function by stretching or compressing the output values by a scale factor of “ $a$ .”

A horizontal dilation transforms the parent function into another function by stretching or compressing the input values by a scale factor of “ $b$ .”

Try it

Transforming Graphs of Quadratics

What transformations and shifts need to be made to change $f (x) = x^{2}$ to $f (x) = - (x - 2)^{2} + 7$ ?

Here is how to determine the transformations needed:

Step 1 - Identify the vertex of the new function. $(2, 7)$

Step 2 - Identify the shifts needed to go from the origin $(0, 0)$ to the new vertex. Right 2, up 7

Step 3 - Ask: Does the parabola open up or down? Does it reflect over the $x$ -axis? Opens down since the equation leads with a negative. Reflects over the $x$ -axis.

The transformations needed are right 2, up 7, and a reflection over the $x$ -axis.

What transformations need to be made to $f (x) = x^{2}$ so it becomes $f (x) = (3 x)^{2}$ ?

Compare your answer:

The needed change impacts the $x$ -values and follows the rule for a horizontal dilation, $f (b x)$ . This dilation represents a horizontal compression by a scale factor of 3.