1. Pre-AP Pre-Calculus Chapter 2, Section 3
Polynomial Functions of Higher Degree with Modeling

2. Polynomial Functions of Higher Degrees & Vocabulary
Cubic Functions –
Quartic Functions –
Term –
Coefficient –
Leading term –

3. Graphical Transformations of Monomial Functions
Describe how the monomial 𝑓 𝑥 = 𝑎 0 𝑥 𝑛 was transferred into the given equation.
𝑔 𝑥 =4 (𝑥+1) 3
ℎ 𝑥 = −(𝑥−2) 4 +5

4. Graph the polynomial function, locate its extrema and zeros.
𝑓 𝑥 = 𝑥 3 +𝑥

5. Graph the polynomial function, locate its extrema and zeros.
𝑓 𝑥 = 𝑥 3 −𝑥

6. Cubic Functions: 2 with positive leading coefficients, 2 with negative leading coefficients

7. Quartic Functions: 2 with positive leading coefficients, 2 with negative leading coefficients

8. Theorem: Local Extrema and Zeros of Polynomial Functions
A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

9. According to the theorem, how many zeros and local extrema could the following functions have?
𝑔 𝑥 =2 𝑥−3 3
𝑓 𝑥 = 𝑥 2 +2
ℎ 𝑥 =8+3𝑥−6 𝑥 4

10. End Behavior Exploration
On the following slides, graph each equation one at a time.
Use the window [-5, 5] by [-15, 15].
Describe the end behavior using lim 𝑥→∞ 𝑓(𝑥) and lim 𝑥→−∞ 𝑓(𝑥)

11. 𝑓 𝑥 = 2𝑥 3
𝑓 𝑥 = −𝑥 3
𝑓 𝑥 = 𝑥 5
𝑓 𝑥 = −0.5𝑥 7

12. 𝑓 𝑥 = −3𝑥 4
𝑓 𝑥 = 0.6𝑥 4
𝑓 𝑥 = 2𝑥 6
𝑓 𝑥 = −0.5𝑥 2

13. 𝑓 𝑥 = −2𝑥 2
𝑓 𝑥 = −0.3𝑥 5
𝑓 𝑥 = 3𝑥 4
𝑓 𝑥 = 2.5𝑥 3

14. Comparing Graphs 𝑦 1 = 𝑥 3 −4 𝑥 2 −5𝑥−3 𝑦 2 = 𝑥 3
Sketch the graph showing both functions
Zoom out till the graphs look nearly identical.
Note the final window

15. Applying Polynomial Theory
Graph the polynomial in a window showing its extrema, zeros, and end behavior.
Describe the end behavior using limits.
𝑓 𝑥 = 𝑥 3 +2 𝑥 2 −11𝑥−12

16. Applying Polynomial Theory
Graph the polynomial in a window showing its extrema, zeros, and end behavior.
Describe the end behavior using limits.
g 𝑥 = 2𝑥 4 +2 𝑥 3 −22 𝑥 2 −18𝑥+35

17. Find the zeros of the function (algebraically)
𝑓 𝑥 = 𝑥 3 − 𝑥 2 −6𝑥

18. Find the zeros of the function (algebraically)
𝑓 𝑥 =3 𝑥 2 +4𝑥−4

19. Factors & Multiplicity
When a factor is repeated, as in 𝑓 𝑥 = (𝑥−2) 3 (𝑥+1) 2 , you can say the polynomial has a repeated zero.
The given function has two repeated zeros: x = _____ and x = ______.
Because the factor (x – 2) occurs three times, the multiplicity of the zero of the function is 3. (it occurs 3 times)
Because the factor (x + 1) occurs twice, the multiplicity of the zero of the function is 2. (it occurs twice)

20. Factors & Multiplicity
State the degree and list the zeros of the polynomial. State the multiplicity of each zero.
𝑓 𝑥 =− 𝑥 3 (𝑥−2)

21. Use the Zoom!! Find all real zeros of
𝑓 𝑥 = 𝑥 𝑥 3 −6.5 𝑥 𝑥−2.4

22. Dixie Packaging Company has contracted to make boxes with a volume of approximately 484 in3 . Squares are to be cut from the corners of a 20-in. by 25-in. piece of cardboard, and the flaps folded up to make an open box. What size squares should be cut from the corners?

23. Factor the given equation
6 𝑥 3 −22 𝑥 2 +12𝑥

24. Highway Safety Division
Stopping Distance
Draw a scatter plot of the data.
Find the quadratic regression model.
Superimpose the regression curve on the graph.
Use the regression model to predict the stopping distance for a vehicle traveling at 25 mpg.
Use the regression model to predict the speed of a car if the stopping distance is 300 ft.
Highway Safety Division
Speed (mph)
Stopping Distance (ft) 10
15.1 20
39.9 30
75.2 40
120.5 50
175.9

26. Ch. 2.3 Homework
Pg : #’s 4, 9, 10,14, 17, 20, 23, 28, 29, 36, 41 (ignore directions about stating whether it crosses the x-axis at corresponding x-axis), 53, 67, 73
(14 total problems)
Gray Book: pages