# 5.9 – Transformations of Polynomial Functions

## Key Terms

• No key terms for this section.

## Review

Graph End Behavior
• Even powers – same end behaviors
• Will have an absolute maximum or minimum
• Odd powers – opposite end behaviors
• Will have relative maximums or minimums, but will also have a true maximum at infinity and a true minimum at negative infinity.
• Positive coefficients – normal end behaviors
• Negative coefficients – opposite end behaviors, reflected (flipped over the x-axis)
• Relative maximums and minimums – odd power polynomials
• The most relative maxima and minima a function can have is one number less that the function’s power.
• The number of a polynomial’s relative extreme values (maxima and minima) is n – 1, where n is the degree (exponent or power) of the polynomial.
• Ex. $f(x)=x^{3}$ can have at most 2 relative maxima or minima.
• Ex. $f(x)=-2(x-3)^{7}$ can have at most 6 relative maxima or minima.
• Absolute maximums and minimums – even power polynomials
• Ex. $f(x)=x^{4}$ can have at most 1 maximum or minimum.
Ex. $f(x)=-2(x-3)^{8}$ can have at most 1 maximum or minimum.

## Notes

Reflections: Flipping
• When you reflect a polynomial over a line, the input (x) and output (y) values change.
• Make a table of x and y values to help you track and graph the changes.
• Flipping over the y = x axis
• Switch all x-values for y-values in the table (and then in the graph).
• Ex. Red (parent function) and Blue (flipped over the y = x axis).

• Flipping over the x-axis
• Multiply the whole function by -1 OUTSIDE the parenthesis.
• Ex. $f(x)=5(x-2)^{2}$ would become $f(x)=-5(x-2)^{2}$
• Ex. $f(x)=x^{4}$ would become $f(x)=-x^{4}$

• Flipping over the y-axis
• Multiply the x-value by -1 INSIDE the parenthesis.
• Ex. $f(x)=5(x-2)^{2}$ would become $f(-x)=5(-x-2)^{2}$
• Ex. $f(x)=x^{5}$ would become $f(x)=(-x)^{5}$

Translating: Shifting
• Horizontal Shift: f(x – k)
• If k > 0, the graph is shifted to the right.
• If k < 0, the graph is shifted to the left.
• Vertical Shift: f(x) + k
• If k > 0, the graph is shifted up.
• If k < 0, the graph is shifted down.
• Ex. $y=x^{4}$ is the parent function (below)

Stretching and Compressing
• To stretch a polynomial function vertically, multiply the function by a number greater than 1.
• Ex. $y=x^{5}$ is the parent function (below)

• To compress a polynomial function vertically, multiply the function by a number between 0 and 1.
Transforming Graphs Chart

## Examples

Shifting from a Shifted Function
• Ex 1. What was the equation of the graph below before it was shifted to the left 1.5 units?
• Where is it now? If it has been shifted left, that means it used to be further to the right.
• Shifting to the right is the same as subtracting a positive number from all x-values.
• If we are currently at (x – 1.5), then we are still right of the origin (which is x minus a positive 1.5).
• We have to subtract a positive 1.5 more to see where we used to be (further to the right).
• So, we have (x – 1.5 – 1.5, which equals x – 3).
• Answer: $G(x)=(x-3)^{3}-(x-3)$

• Ex 2. The graph of g(x), shown below, resembles the graph of $f(x)=x^{4}-x^{2}$, but it has been changed somewhat. Which of the following could be the equation of g(x)?
• Since it has shifted down by 2, the answer is: $f(x)=x^{4}-x^{2}-2$

• Ex 3. Which rule should be applied to reflect $f(x)=x^{3}$ over the y-axis?
• Flipping over the y-axis involves changing the sign of the x-value; so…
• Answer: substitute -x for x, then simplify f(-x).
• Ex 4. What is the equation of this function after it is reflected over the x-axis?
• Since the graph has a negative coefficient (see the -3).  Change it to a positive 3.
• Answer: $f(x)=3(x+2)^{3}$

• Ex 5. What is the equation of the graph that represents the parent function $f(x^{4})$ stretched vertically by a factor of 2, and then shifted left 3 spaces.
• Stretching affects the coefficient of the x-values. Shifting left subtracts a negative from the x-values.
• Remember: subtracting a negative is the same as adding a positive.
• Answer: $g(x)=2(x+3)^{4}$
• Ex 6. Which function results after applying the sequence of transformations to $f(x)=x^{5}$?
1. Reflect over the y-axis
2. Shift left 1 unit
3. Vertically compress by $\frac{1}{3}$
• Answer: $f(x)=\frac{1}{3}(-x+1)^{5}$
• Ex 7. What rule should be used to transform a table of data to represent the reflection of f(x) over the line y = x?
• Answer: Switch the x-values and y-values in the table.

• Ex 8. What are the end behaviors of $f(x)=-2(x-17)^{4}$?
• Since the leading coefficient is negative, the graph is flipped.
• Since the exponent is even, both of the ends do the same thing.
• So, the answer is: “both ends go down”
• Ex 9. Choose the characteristics of the function: $f(x)=-(x+4)^{5}$?
• Its parent function was reflected and translated left.
• The left end goes up and the right end goes down.
• There are 5 roots and at most 4 relative extrema.
• Ex 10. Which of the two functions below has the smallest minimum y-value?
• $f(x)=(x-13)^{4}-2$ and $g(x)=3x^{3}+2$
• f(x) has a minimum of -2
• g(x) has a minimum of negative infinity