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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 xaxis)
 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. can have at most 2 relative maxima or minima.
 Ex. can have at most 6 relative maxima or minima.
 Absolute maximums and minimums – even power polynomials
 Ex. can have at most 1 maximum or minimum.
Ex. can have at most 1 maximum or minimum.

Notes
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. is the parent function (below)

Stretching and Compressing 
 To stretch a polynomial function vertically, multiply the function by a number greater than 1.
 Ex. 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 xvalues.
 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:

 Ex 2. The graph of g(x), shown below, resembles the graph of , 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:

 Ex 3. Which rule should be applied to reflect over the yaxis?
 Flipping over the yaxis involves changing the sign of the xvalue; 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 xaxis?
 Since the graph has a negative coefficient (see the 3). Change it to a positive 3.
 Answer:

 Ex 5. What is the equation of the graph that represents the parent function stretched vertically by a factor of 2, and then shifted left 3 spaces.
 Stretching affects the coefficient of the xvalues. Shifting left subtracts a negative from the xvalues.
 Remember: subtracting a negative is the same as adding a positive.
 Answer:

 Ex 6. Which function results after applying the sequence of transformations to ?
 Reflect over the yaxis
 Shift left 1 unit
 Vertically compress by
 Answer:

 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 xvalues and yvalues in the table.

 Ex 8. What are the end behaviors of ?
 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: ?
 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 yvalue?
 and
 f(x) has a minimum of 2
 g(x) has a minimum of negative infinity

Important!
Practice (Apex Study 5.9)
 Try practice problems on Pgs 14, 16, 22
 Mandatory: write and answer problems on Pgs 15, 23
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