Objectives
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Apply transformations including horizontal and vertical shifts, stretches, and reflections to a function. Express the result of these transformations graphically and algebraically.
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.
(a)
How is the graph of \(f(x)+1\) related to that of \(f(x)\text{?}\)
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Shifted up \(1\) unit
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Shifted left \(1\) unit
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Shifted down \(1\) unit
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Shifted right \(1\) unit
Activity 2.4.3.
(a)
How is the graph of \(f(x)-2\) related to that of \(f(x)\text{?}\)
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Shifted up \(2\) units
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Shifted left \(2\) units
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Shifted down \(2\) units
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Shifted right \(2\) units
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.
(a)
How is the graph of \(f(x+1)\) related to that of \(f(x)\text{?}\)
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Shifted up by 1 unit
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Shifted left 1 unit
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Shifted down 1 unit
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Shifted right 1 unit
Activity 2.4.7.
(a)
How is the graph of \(f(x-2)\) related to that of \(f(x)\text{?}\)
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Shifted up by 2 units
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Shifted left 2 units
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Shifted down 2 units
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Shifted right 2 units
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)
\(f(x-4)+1\)
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Shifted down \(4\) units
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Shifted left \(4\) units
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Shifted down \(1\) unit
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Shifted right \(4\) units
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Shifted up \(1\) unit
(b)
\(f(x+3)-2\)
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Shifted down \(2\) units
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Shifted left \(3\) units
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Shifted up \(3\) units
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Shifted right \(3\) units
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Shifted up \(2\) units
Activity 2.4.11.
For each of the following, use the information given to find another point on the graph.
(a)
If \((2,3)\) is a point on the graph of \(f(x)\text{,}\) what point must be on the graph of \(f(x)+2\text{?}\)
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\(\displaystyle (4,3)\)
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\(\displaystyle (2,5)\)
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\(\displaystyle (4,5)\)
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\(\displaystyle (2,1)\)
(b)
If \((-1,6)\) is a point on the graph of \(g(x)\text{,}\) what point must be on the graph of \(g(x-4)\text{?}\)
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\(\displaystyle (-1,10)\)
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\(\displaystyle (3,6)\)
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\(\displaystyle (-1,2)\)
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\(\displaystyle (-5,6)\)
(c)
If \((-2,-5)\) is a point on the graph of \(h(x)\text{,}\) what point must be on the graph of \(h(x+1)-5\text{?}\)
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\(\displaystyle (-3,-10)\)
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\(\displaystyle (-3,0)\)
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\(\displaystyle (-1,-10)\)
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\(\displaystyle (-1,0)\)
Activity 2.4.12.
(a)
How is the graph of \(-f(x)\) related to that of \(f(x)\text{?}\)
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Shifted down \(2\) units
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted right \(2\) units
Activity 2.4.13.
(a)
How is the graph of \(f(-x)\) related to that of \(f(x)\text{?}\)
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Shifted down \(2\) units
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted left \(2\) units
Remark 2.4.14.
Notice that in Activity 2.4.12, the \(y\)-values of the transformed graph are changed while the \(x\)-values remain the same. While in Activity 2.4.13, the \(x\)-values of the transformed graph are changed while the \(y\)-values remain the same.
Definition 2.4.15.
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 2.4.16.
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 2.4.17.
Consider the following graph of the function \(f(x)\text{.}\)
Diagram Exploration Keyboard Controls
(a)
How is the graph of \(-f(x+2)+3\) related to that of \(f(x)\text{?}\)
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Shifted up \(2\) units
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Shifted up \(3\) units
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted left \(3\) units
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Shifted left \(2\) units
(b)
Which of the following represents the graph of the transformed function \(g(x)=-f(x+2)+3\text{?}\)
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Diagram Exploration Keyboard Controls
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Diagram Exploration Keyboard Controls
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Diagram Exploration Keyboard Controls
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Diagram Exploration Keyboard Controls
Remark 2.4.18.
Notice that in Activity 2.4.17 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 2.4.19.
(a)
How is the graph of \(g(x)\) related to that of \(f(x)\text{?}\)
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Shifted up \(3\) units
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Shifted up \(1\) unit
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted left \(1\) unit
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Shifted right \(4\) units
(b)
List the order the transformations must be applied.
(c)
Write an equation for the graphed function \(g(x)\) using transformations of the graph \(f(x)\text{.}\)
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\(\displaystyle g(x)=-f(x)+3\)
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\(\displaystyle g(x)=f(-x)+3 \)
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\(\displaystyle g(x)=f(-x+3) \)
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\(\displaystyle g(x)=-f(x+3) \)
Activity 2.4.20.
For each of the following, use the information given to find another point on the graph.
(a)
If \((1,6)\) is a point on the graph of \(f(x)\text{,}\) what point must be on the graph of \(-f(x-2)\text{?}\)
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\(\displaystyle (-3,6)\)
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\(\displaystyle (-1,6)\)
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\(\displaystyle (-1,4)\)
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\(\displaystyle (3,-6)\)
(b)
If \((-2,-4)\) is a point on the graph of \(g(x)\text{,}\) what point must be on the graph of \(g(-x)+3\text{?}\)
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\(\displaystyle (-2,-7)\)
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\(\displaystyle (2,-1)\)
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\(\displaystyle (2,-7)\)
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\(\displaystyle (-2,7)\)
Activity 2.4.21.
(a)
Consider the \(y\)-value of the two graphs at \(x=1\text{.}\) How do they compare?
(b)
How is the graph of \(2f(x)\) related to that of \(f(x)\text{?}\)
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Vertically stretched by a factor of \(2\)
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Vertically compressed by a factor of \(2\)
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Horizontally stretched by a factor of \(2\)
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Horizontally compressed by a factor of \(2\)
Activity 2.4.22.
(a)
Consider an \(x\)-value of the two graphs where \(y=1\text{.}\) How do they compare?
(b)
How is the graph of \(f(2x)\) related to that of \(f(x)\text{?}\)
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Vertically stretched by a factor of \(2\)
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Vertically compressed by a factor of \(2\)
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Horizontally stretched by a factor of \(2\)
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Horizontally compressed by a factor of \(2\)
Remark 2.4.23.
Notice that in Activity 2.4.21 the \(y\)-values are doubled while the \(x\)-values remain the same. While, in Activity 2.4.22 the \(x\)-values are cut in half while the \(y\)-values remain the same.
Definition 2.4.24.
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 2.4.25.
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 2.4.26.
We often use a set of basic functions with which to begin transformations. We call these parent functions. A library of these is collected in Section A.1, but here are a few of the most common ones.
Diagram Exploration Keyboard Controls
Activity 2.4.33.
Consider the function \(g(x)=3\sqrt{-x}+2\)
(a)
Identify the parent function \(f(x)\text{.}\)
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\(\displaystyle f(x)=x^{2}\)
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\(\displaystyle f(x)=\lvert x \rvert\)
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\(\displaystyle f(x)=\sqrt{x}\)
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\(\displaystyle f(x)=x\)
(b)
Graph the parent function \(f(x)\text{.}\)
(c)
How is the graph of \(g(x)\) related to that of the parent function\(f(x)\text{?}\)
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted down \(2\) units
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Shifted up \(2\) units
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Vertically stretched by a factor of \(3\)
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Horizontally compressed by a factor of \(3\)
(d)
Graph the transformed function \(g(x)\text{.}\)
Activity 2.4.34.
Consider the following graph of the function \(g(x)\text{.}\)
Diagram Exploration Keyboard Controls
(a)
Identify the parent function.
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\(\displaystyle f(x)=x^{2}\)
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\(\displaystyle f(x)=\lvert x \rvert\)
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\(\displaystyle f(x)=\sqrt{x}\)
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\(\displaystyle f(x)=x\)
(b)
How is the graph of \(g(x)\) related to that of the parent function\(f(x)\text{?}\)
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Reflected over the \(x\)-axis
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Reflected over the \(y\)-axis
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Shifted down \(3\) units
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Shifted up \(3\) units
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Shifted left \(2\) units
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Shifted right \(2\) units
(c)
Write an equation to represent the transformed function \(g(x)\text{.}\)
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\(\displaystyle g(x)=-(x-2)^{2}-3\)
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\(\displaystyle g(x)=-(x+2)^{2}+3\)
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\(\displaystyle g(x)=(-x+2)^{2}-3\)
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\(\displaystyle g(x)=-(x+2)^{2}-3\)
Activity 2.4.35.
For each of the following, write a formula for the new function \(g(x)\) when the graph of \(f(x)\) is transformed as described.
(a)
\(f(x) = |x|\) is shifted \(3\) units down
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\(\displaystyle g(x)=|x-3|\)
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\(\displaystyle g(x)=|x|+3\)
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\(\displaystyle g(x)=|x|-3\)
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\(\displaystyle g(x)=|x+3|\)
(b)
\(f(x) = x^2\) is shifted \(2\) units left and reflected over the \(x\)-axis
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\(\displaystyle h(x)= -\sqrt{x-2}\)
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\(\displaystyle h(x)=-(x-2)^2\)
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\(\displaystyle h(x)=\sqrt{-x+2}\)
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\(\displaystyle h(x)=-(x+2)^2\)
(c)
\(f(x) = x^3\) is reflected over the \(y\)-axis, vertically stretched by a factor of \(2\text{,}\) and shifted \(1\) unit up
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\(\displaystyle g(x)= 2(-x)^3+1\)
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\(\displaystyle g(x)=(-2x)^3-1\)
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\(\displaystyle g(x)=-(2x)^3+1\)
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\(\displaystyle g(x)-2x^3+1\)
Subsection 2.4.2 Exercises
Exercises available at
https://tbil.org/preview/precalculus/exercises/#/bank/FN4/
