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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 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).

Alg2B 5.9 Flip y equals x

Alg2B 5.9 Flip y equals x ex2

  • 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}

Alg2B 5.9 Flip x-axis

  • 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}

Alg2B 5.9 Flip y-axis

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)

Alg2B 5.9 Shift Up

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)

Alg2B 5.9 Stretching

  • To compress a polynomial function vertically, multiply the function by a number between 0 and 1.
Transforming Graphs Chart
Alg2B 5.9 Graph End Behavior

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)

Alg2B 5.9 Q1Ex1

  • 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

Alg2B 5.9 Q1Ex2

  • 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}

Alg2B 5.9 Q1Ex4

  • 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.

Alg2B 5.9 Q1Ex7

  • 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
        • Answer: g(x)

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