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Precalculus for Team-Based Inquiry Learning: 2024 Development Edition — Instructor Version

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Section 2.4 Transformation of Functions (FN4)

Subsection 2.4.1 Activities

Remark 2.4.1.

Informally, a transformation of a given function is an algebraic process by which we change the function to a related function that has the same fundamental shape, but may be shifted, reflected, and/or stretched in a systematic way.

Activity 2.4.2.

Consider the following two graphs.
(a)
How is the graph of \(f(x)+1\) related to that of \(f(x)\text{?}\)
  1. Shifted up \(1\) unit
  2. Shifted left \(1\) unit
  3. Shifted down \(1\) unit
  4. Shifted right \(1\) unit
Answer.
A

Activity 2.4.3.

Consider the following two graphs.
(a)
How is the graph of \(f(x)-2\) related to that of \(f(x)\text{?}\)
  1. Shifted up \(2\) units
  2. Shifted left \(2\) units
  3. Shifted down \(2\) units
  4. Shifted right \(2\) units
Answer.
C

Remark 2.4.4.

Notice that in Activity 2.4.2 and Activity 2.4.3, the \(y\)-values of the transformed graph are changed while the \(x\)-values remain the same.

Definition 2.4.5.

Given a function \(f(x)\) and a constant \(c\text{,}\) the transformed function \(g(x)=f(x)+c\) is a vertical translation of the graph of \(f(x)\text{.}\) That is, all the outputs change by \(c\) units. If \(c\) is positive, the graph will shift up. If \(c\) is negative, the graph will shift down.

Activity 2.4.6.

Consider the following two graphs.
(a)
How is the graph of \(f(x+1)\) related to that of \(f(x)\text{?}\)
  1. Shifted up by 1 unit
  2. Shifted left 1 unit
  3. Shifted down 1 unit
  4. Shifted right 1 unit
Answer.
B

Activity 2.4.7.

Consider the following two graphs.
(a)
How is the graph of \(f(x-3)\) related to that of \(f(x)\text{?}\)
  1. Shifted up by 3 units
  2. Shifted left 3 units
  3. Shifted down 3 units
  4. Shifted right 3 units
Answer.
D

Remark 2.4.8.

Notice that in Activity 2.4.6 and Activity 2.4.7, the \(x\)-values of the transformed graph are changed while the \(y\)-values remain the same.

Definition 2.4.9.

Given a function \(f(x)\) and a constant \(c\text{,}\) the transformed function \(g(x)=f(x+c)\) is a horizontal translation of the graph of \(f(x)\text{.}\) If \(c\) is positive, the graph will shift left. If \(c\) is negative, the graph will shift right.

Activity 2.4.10.

Describe how the graph of the function is a transformation of the graph of the original function \(f\text{.}\)
(a)
\(f(x-4)+1\)
  1. Shifted down \(4\) units
  2. Shifted left \(4\) units
  3. Shifted down \(1\) unit
  4. Shifted right \(4\) units
  5. Shifted up \(1\) unit
Answer.
D and E
(b)
\(f(x+3)-2\)
  1. Shifted down \(2\) units
  2. Shifted left \(3\) units
  3. Shifted up \(3\) unit
  4. Shifted right \(3\) units
  5. Shifted up \(2\) unit
Answer.
A and B

Activity 2.4.11.

Consider the following two graphs.
(a)
How is the graph of \(-f(x)\) related to that of \(f(x)\text{?}\)
  1. Shifted down \(2\) units
  2. Reflected over the \(x\)-axis
  3. Reflected over the \(y\)-axis
  4. Shifted right \(2\) units
Answer.
B

Activity 2.4.12.

Consider the following two graphs.
(a)
How is the graph of \(f(-x)\) related to that of \(f(x)\text{?}\)
  1. Shifted down \(2\) units
  2. Reflected over the \(x\)-axis
  3. Reflected over the \(y\)-axis
  4. Shifted left \(2\) units
Answer.
C

Remark 2.4.13.

Notice that in Activity 2.4.11, the \(y\)-values of the transformed graph are changed while the \(x\)-values remain the same. While in Activity 2.4.12, the \(x\)-values of the transformed graph are changed while the \(y\)-values remain the same.

Definition 2.4.14.

Given a function \(f(x)\text{,}\) the transformed function \(g(x)=-f(x)\) is a vertical reflection of the graph of \(f(x)\text{.}\) That is, all the outputs are multiplied by \(-1\text{.}\) The new graph is a reflection of the old graph about the \(x\)-axis.

Definition 2.4.15.

Given a function \(f(x)\text{,}\) the transformed function \(y=g(x)=f(-x)\) is a horizontal reflection of the graph of \(f(x)\text{.}\) That is, all the inputs are multiplied by \(-1\text{.}\) The new graph is a reflection of the old graph about the \(y\)-axis.

Activity 2.4.16.

Consider the following graph of the function \(f(x)\text{.}\)
(a)
How is the graph of \(-f(x+2)+3\) related to that of \(f(x)\text{?}\)
  1. Shifted up \(2\) units
  2. Shifted up \(3\) units
  3. Reflected over the \(x\)-axis
  4. Reflected over the \(y\)-axis
  5. Shifted left \(3\) units
  6. Shifted left \(2\) units
Answer.
B, C and F
(b)
Which of the following represents the graph of the transformed function \(g(x)=-f(x+2)+3\text{?}\)
Answer.
A

Remark 2.4.17.

Notice that in Activity 2.4.16 the resulting graph is different if you perform the reflection first and then the vertical shift, versus the other order. When combining transformations, it is very important to consider the order of the transformations. Be sure to follow the order of operations.

Activity 2.4.18.

Consider the following two graphs.
(a)
How is the graph of \(g(x)\) related to that of \(f(x)\text{?}\)
  1. Shifted up \(3\) units
  2. Shifted up \(1\) unit
  3. Reflected over the \(x\)-axis
  4. Reflected over the \(y\)-axis
  5. Shifted left \(1\) unit
  6. Shifted right \(4\) units
Answer.
A and D
(b)
List the order the transformations must be applied.
Answer.
Reflect over the \(y\)-axis and then shift up \(4\) units.
(c)
Write an equation for the graphed function \(g(x)\) using transformations of the graph \(f(x)\text{.}\)
  1. \(\displaystyle g(x)=-f(x)+3\)
  2. \(\displaystyle g(x)=f(-x)+3 \)
  3. \(\displaystyle g(x)=f(-x+3) \)
  4. \(\displaystyle g(x)=-f(x+3) \)
Answer.
B

Activity 2.4.19.

Consider the following two graphs.
(a)
Consider the \(y\)-value of the two graphs at \(x=1\text{.}\) How do they compare?
  1. The \(y\)-value of \(2f(x)\) is twice that of \(f(x)\text{.}\)
  2. The \(y\)-value of \(2f(x)\) is half that of \(f(x)\text{.}\)
  3. The \(y\)-value of \(2f(x)\) and \(f(x)\) are the same.
  4. The \(y\)-value of \(2f(x)\) is negative that of \(f(x)\text{.}\)
Answer.
A
(b)
How is the graph of \(2f(x)\) related to that of \(f(x)\text{?}\)
  1. Vertically stretched by a factor of \(2\)
  2. Vertically compressed by a factor of \(2\)
  3. Horizontally stretched by a factor of \(2\)
  4. Horizontally compressed by a factor of \(2\)
Answer.
A

Activity 2.4.20.

Consider the following two graphs.
(a)
Consider a \(x\)-value of the two graphs at \(y=1\text{.}\) How do they compare?
  1. The \(x\)-value of \(2f(x)\) is twice that of \(f(x)\text{.}\)
  2. The \(x\)-value of \(2f(x)\) is half that of \(f(x)\text{.}\)
  3. The \(x\)-value of \(2f(x)\) and \(f(x)\) are the same.
  4. The \(x\)-value of \(2f(x)\) is negative that of \(f(x)\text{.}\)
Answer.
B
(b)
How is the graph of \(f(2x)\) related to that of \(f(x)\text{?}\)
  1. Vertically stretched by a factor of \(2\)
  2. Vertically compressed by a factor of \(2\)
  3. Horizontally stretched by a factor of \(2\)
  4. Horizontally compressed by a factor of \(2\)
Answer.
D

Remark 2.4.21.

Notice that in Activity 2.4.19 the \(y\)-values are doubled while the \(x\)-values remain the same. While, in Activity 2.4.20 the \(x\)-values are cut in half while the \(y\)-values remain the same.

Definition 2.4.22.

Given a function \(f(x)\text{,}\) the transformed function \(g(x)=af(x)\) is a vertical stretch or vertical compression of the graph of \(f(x)\text{.}\) That is, all the outputs are multiplied by \(a\text{.}\) If \(a \gt 1\text{,}\) the new graph is a vertical stretch of the old graph away from the \(x\)-axis. If \(0 \lt a \lt 1\text{,}\) the new graph is a vertical compression of the old graph towards the \(x\)-axis. Points on the \(x\)-axis are unchanged.

Definition 2.4.23.

Given a function \(f(x)\text{,}\) the transformed function \(g(x)=f(ax)\) is a horizontal stretch or horizontal compression of the graph of \(f(x)\text{.}\) That is, all the inputs are divided by \(a\text{.}\) If \(a \gt 1\text{,}\) the new graph is a horizontal compression of the old graph toward the \(y\)-axis. If \(0 \lt a \lt 1\text{,}\) the new graph is a horizontal stretch of the old graph away from the \(y\)-axis. Points on the \(y\)-axis are unchanged.

Remark 2.4.24.

We often use a set of basic functions with which to begin transformations. We call these parent functions.

Activity 2.4.25.

Consider the function \(g(x)=3\sqrt{-x}+2\)
(a)
Identify the parent function \(f(x)\text{.}\)
  1. \(\displaystyle f(x)=x^{2}\)
  2. \(\displaystyle f(x)=\lvert x \rvert\)
  3. \(\displaystyle f(x)=\sqrt{x}\)
  4. \(\displaystyle f(x)=x\)
Answer.
C
(b)
Graph the parent function \(f(x)\text{.}\)
Answer.
(c)
How is the graph of \(g(x)\) related to that of the parent function\(f(x)\text{?}\)
  1. Reflected over the \(x\)-axis
  2. Reflected over the \(y\)-axis
  3. Shifted down \(2\) units
  4. Shifted up \(2\) units
  5. Vertically stretched by a factor of \(3\)
  6. Horizontally compressed by a factor of \(3\)
Answer.
B, D, and E
(d)
Graph the transformed function \(g(x)\text{.}\)
Answer.

Activity 2.4.26.

Consider the following graph of the function \(g(x)\text{.}\)
(a)
Identify the parent function.
  1. \(\displaystyle f(x)=x^{2}\)
  2. \(\displaystyle f(x)=\lvert x \rvert\)
  3. \(\displaystyle f(x)=\sqrt{x}\)
  4. \(\displaystyle f(x)=x\)
Answer.
A
(b)
How is the graph of \(g(x)\) related to that of the parent function\(f(x)\text{?}\)
  1. Reflected over the \(x\)-axis
  2. Reflected over the \(y\)-axis
  3. Shifted down \(3\) units
  4. Shifted up \(3\) units
  5. Shifted left \(2\) units
  6. Shifted right \(2\) units
Answer.
A, B, and D
(c)
Write an equation to represent the transformed function \(g(x)\text{.}\)
  1. \(\displaystyle g(x)=-(x-2)^{2}-3\)
  2. \(\displaystyle g(x)=-(x+2)^{2}+3\)
  3. \(\displaystyle g(x)=(-x+2)^{2}-3\)
  4. \(\displaystyle g(x)=-(x+2)^{2}-3\)
Answer.
D

Subsection 2.4.2 Videos

It would be great to include videos down here, like in the Calculus book!
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