Skip to main content

Lab Manual for MTH 111: Precalculus I: Functions

Mathematics Faculty

Section 1.6 Graph Transformations and Symmetry

In this section, we’ll look at ways we can algebraically manipulate the inputs and outputs of a function and see the impact this has on the overall function and its graph. We’ll also look at the symmetry that some functions have related and how this symmetry relates to some of the transformations we’ll see.

Subsection Textbook Reference

This relates to content in §3.5 of Algebra and Trigonometry 2e
 1 
http://tiny.cc/111Z-Textbook
.

Exercises Preparation Exercises

1.

Suppose the following graph represents the function \(A\) which gives the air temperature (in °F above 65°F) in a classroom \(t\) hours after midnight each weekday.
described in detail following the image
Graph of a function depicting air temperature in a classroom. The horizontal axis is labeled "\(t\text{,}\) hours after midnight" and is shown for \(t\) between 0 and 24. The vertical axis is labeled "\(y\text{,}\) degrees above 65°F" and is shown from -6 to 6. The graph begins at \((0,-3)\text{,}\) is constant until the point \((5,-3)\text{,}\) increases linearly to the point \((7,2)\text{,}\) is constant until the point \((20,2)\text{,}\) then decreases linearly to the point \((24,-3)\text{.}\)
Figure 1.6.1. \(y=A(t)\)
(a)
Let \(W\) be a new function that has the same schedule as the room used by \(A\text{,}\) but is for a room that is always two degrees warmer than the room used by \(A\text{.}\) In Figure 1.6.1, sketch the graph of the function \(W\text{.}\)
(b)
What expression could we use for \(W\) to represent the function used to create the graph for the room that is always two degrees warmer than the room used for \(A\text{?}\)
(c)
If we were to vertically shift the graph of \(y=A(t)\) down by one unit in the \(y\) direction to create a new function \(C\text{,}\) what would that mean in the context of the temperature of the room and what expression could we use to represent \(C\text{?}\)

Exercises Practice Exercises

1.

Let \(f\) be a function.
(a)
If \(k_1(x) = -f(x+3)\text{,}\) state the transformations that take the graph of \(y=f(x)\) to the graph of \(y=k_1(x)\text{.}\)
(b)
If \(k_2(x) = f(\frac{1}{5}x)+3\text{,}\) state the transformations that take the graph of \(y=f(x)\) to the graph of \(y=k_2(x)\text{.}\)

2.

Let \(g\) be a function for which we know that \(g(3) = 5\text{.}\)
(a)
If \(m(x) = -2g(x-4)+7\text{,}\) state a sequence of transformations that takes the graph of \(y=g(x)\) to the graph of \(y=m(x)\text{?}\)
(b)
What point do you know is on the graph of \(y=g(x)\text{?}\)
(c)
What point do you know is on the graph of \(y=m(x)\text{?}\)

3.

Given in Figure 1.6.2 is the graph of \(y = f(x)\text{.}\)
described in detail following the image
Graph of a curve passing smoothly through the points \((-6,3)\text{;}\) \((-3,-1)\text{;}\) \((0,0)\text{;}\) \((3,1)\text{;}\) and \((6,-3)\text{.}\) If you rotated the graph by 180°, it would appear the same.
Figure 1.6.2. \(y=f(x)\)
(a)
Based on the graph, would you say that \(f\) is even, odd, or neither? Explain your answer.

4.

(a)
Algebraically determine if the function \(g(x) = -2x^2 - 3x\) is even, odd, or neither.

5.

(a)
If \(k_3(x) = 4\cdot f(6x-2)+5\text{,}\) state the transformations that take the graph of \(y=f(x)\) to the graph of \(y=k_3(x)\text{.}\)
(b)
If \(k_4(x) = \frac{7}{3}\cdot f(\frac{1}{5}x-1)\text{,}\) state the transformations that take the graph of \(y=f(x)\) to the graph of \(y=k_4(x)\text{.}\)

Subsection Definitions

Definition 1.6.3. Vertical Shift.

Given a function \(f\text{,}\) the graph of \(g(x) = f(x) + k\) for some real number \(k\) is a vertical shift of the graph of \(y = f(x)\text{.}\)
  • If \(k \gt 0\text{,}\) \(g\) will be the graph of \(f\) shifted up by \(k\) units.
  • If \(k \lt 0\text{,}\) \(g\) will be the graph of \(f\) shifted down by \(k\) units.

Example 1.6.4.

View this Desmos graph
 2 
https://tiny.cc/111Z-VertShift
to see an interactive example of the definition.

Definition 1.6.5. Horizontal Shift.

Given a function \(f\text{,}\) the graph of \(g(x) = f(x-h)\) for some real number \(h\) is a horizontal shift of the graph of \(y = f(x)\text{.}\)
  • If \(h \gt 0\text{,}\) \(g\) will be the graph of \(f\) shifted right by \(h\) units.
  • If \(h \lt 0\text{,}\) \(g\) will be the graph of \(f\) shifted left by \(h\) units.

Example 1.6.6.

View this Desmos graph
 3 
https://tiny.cc/111Z-HorizShift
to see an interactive example of the definition.

Definition 1.6.7. Vertical Stretch/Compression.

Given a function \(f\text{,}\) the graph of \(g(x) = a \cdot f(x)\) for some real number \(a\text{,}\) where \(a \neq 0\text{,}\) is a vertical stretch or vertical compression of the graph of \(y = f(x)\text{.}\)
  • If \(a \gt 1\text{,}\) \(g\) will be the graph of \(f\) vertically stretched by a factor of \(a\text{.}\)
  • If \(0 \lt a \lt 0\text{,}\) \(g\) will be the graph of \(f\) vertically compressed by a factor of \(a\text{.}\)
  • If \(a \lt 0\text{,}\) \(g\) will be a combination of a vertical reflection over the \(x\)-axis and a vertical stretch or compression of the graph of \(f\text{.}\)

Example 1.6.8.

View this Desmos graph
 4 
https://tiny.cc/111Z-VertStretch
to see an interactive example of the definition.

Definition 1.6.9. Horizontal Stretch/Compression.

Given a function \(f\text{,}\) the graph of \(g(x) = f(b \cdot x)\) for some real number \(b\text{,}\) where \(b \neq 0\text{,}\) is a horizontal stretch or horizontal compression of the graph of \(y = f(x)\text{.}\)
  • If \(b \gt 1\text{,}\) \(g\) will be the graph of \(f\) horizontally compressed by a factor of \(\frac{1}{b}\text{.}\)
  • If \(0 \lt b \lt 0\text{,}\) \(g\) will be the graph of \(f\) horizontally stretched by a factor of \(\frac{1}{b}\text{.}\)
  • If \(b \lt 0\text{,}\) \(g\) will be a combination of a horizontal reflection over the \(y\)-axis and a horizontal stretch or compression of the graph of \(f\text{.}\)

Example 1.6.10.

View this Desmos graph
 5 
https://tiny.cc/111Z-HorizStretch
to see an interactive example of the definition.

Definition 1.6.11. Vertical Reflection.

Given a function \(f\text{,}\) the graph of \(y = -f(x)\) is a vertical reflection of the graph of \(y = f(x)\) over the \(x\)-axis.

Example 1.6.12.

View this Desmos graph
 6 
https://tiny.cc/111Z-VertReflect
to see an interactive example of the definition.

Definition 1.6.13. Horizontal Reflection.

Given a function \(f\text{,}\) the graph of \(y = f(-x)\) is a horizontal reflection of the graph of \(y = f(x)\) over the \(y\)-axis.

Example 1.6.14.

View this Desmos graph
 7 
https://tiny.cc/111Z-HorizReflect
to see an interactive example of the definition.

Definition 1.6.15. Combined Transformations.

Given a function \(f\text{,}\) the combined vertical transformations written in the form \(y = a \cdot f(x) + k, a \neq 0\text{,}\) would be applied in the order:
  1. A vertical reflection over the \(x\)-axis, if \(a \lt 0\)
  2. A vertical stretch or compression by a factor of \(|a|\)
  3. A vertical shift up or down by \(k\) units
Given a function \(f\text{,}\) the combined horizontal transformations written in the form \(y = f(b \cdot (x - h)), b \neq 0\text{,}\) would be applied in the order:
  1. A horizontal reflection over the \(y\)-axis, if \(b \lt 0\)
  2. A horizontal stretch or compression by a factor of \(|\frac{1}{b}|\)
  3. A horizontal shift left or right by \(h\) units
Given a function \(f\text{,}\) the combined horizontal transformations written in the form \(y = f(bx - h), b \neq 0\text{,}\) would be applied in the order:
  1. A horizontal shift left or right by \(h\) units
  2. A horizontal reflection over the \(y\)-axis, if \(b \lt 0\)
  3. A horizontal stretch or compression by a factor of \(|\frac{1}{b}|\)

Definition 1.6.16. Even Function.

Given a function \(f\text{,}\) if \(f(-x) = f(x)\) for every input \(x\text{,}\) then \(f\) is an even function. We describe even functions as being symmetrical about the \(y\)-axis.

Example 1.6.17.

View this Desmos graph
 8 
https://tiny.cc/111Z-EvenFunction
to see an interactive example of the definition.

Definition 1.6.18. Odd Function.

Given a function \(f\text{,}\) if \(-f(-x) = f(x)\) for every input \(x\text{,}\) then \(f\) is an odd function. We describe odd functions as being symmetrical about the origin. Note: \(f(x) = -f(-x)\) is equivalent to the statement \(f(-x) = -f(x)\text{.}\)

Example 1.6.19.

View this Desmos graph
 9 
https://tiny.cc/111Z-OddFunction
to see an interactive example of the definition.

Exercises Exercises

1.

(a)
What is meant by an "inside" change? How do inside changes impact the graph of a function?
(b)
What is meant by an "outside" change? How do outside changes impact the graph of a function?
(c)
What is the relationship between even and odd functions and transformations?

2.

\(f\) is a function and \(f(-20) = 32\text{.}\)
(a)
If \(g(x) = -\frac{1}{8} f(-2x+10)+5\text{,}\) list the sequence of transformations that take the graph of \(y=f(x)\) to the graph of \(y=g(x)\text{.}\)
(b)
What point is on the graph of \(y=f(x)\) and what point will be on the graph of \(y=g(x)\text{?}\)

Reflection Reflection

1.

On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
PrevTopNext