1. 2.8 Analyze Graphs of Polynomial Functions

2. Example 1
Graph the function. Identify the x-intercepts (zeors), the points where the local maximums and local minimums occur, and the turning points of the function.
h (x) = 0.5x3 + x2 – x + 2
x-intercept: –3.074 local minimum: (0.387, 1.792) local maximum: (–1.721, 4.134)
2 turning points

3. x-intercepts: 1, 4 local minimum: (3.25, –17.056) local maximum: none
Example 2
Graph the function. Identify the x-intercepts (zeros), the points where the local maximums and local minimums occur, and the turning points of the function.
x-intercepts: 1, 4 local minimum: (3.25, –17.056) local maximum: none
2 turning points

4. Turning Points of a Polynomial Function
The graph of every polynomial function of degree n has at most n-1 turning points.
If a polynomial function has n distinct real zeros, then its graph has exactly n-1 turning points.

5. 1. Multiply (x + 2)(3x + 1)
ANSWER
3x2 + 7x + 2
2. Find the intercepts of y = (x + 6)(x – 5)
ANSWER
–6 and 5

6. 3. An object is projected vertically upward. Its distance
3. An object is projected vertically upward. Its distance D in feet above the ground after t seconds is given by D = –16t t Find its maximum distance above the ground. +
ANSWER
424 ft

7. EXAMPLE 1
Use x-intercepts to graph a polynomial function
Graph the function f (x) = (x + 3)(x – 2)2. 1 6
SOLUTION
STEP 1
Plot: the intercepts. Because –3 and 2 are
zeros of f, plot (–3, 0) and (2, 0).
STEP 2
Plot: points between and beyond the x-intercepts.

8. EXAMPLE 1
Use x-intercepts to graph a polynomial function
STEP 3 1 6
Determine: end behavior. Because f has three factors of the form x – k and a constant factor of , it is a cubic function
with a positive leading coefficient. So, f (x) → – ∞ as x → – ∞ and f (x) → + ∞ as x → + ∞.
STEP 4
Draw the graph so that it passes through the plotted points and has the appropriate end behavior.

9. EXAMPLE 2
Find turning points
Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.
a. f (x) = x3 – 3x2 + 6
b. g (x) 5 x4 – 6x3 + 3x2 + 10x – 3

10. EXAMPLE 2
Find turning points
SOLUTION
a. f (x) = x3 – 3x2 + 6
a. Use a graphing calculator to graph the function.
Notice that the graph of f has one x-intercept and two turning points.
You can use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.
The x-intercept of the graph is x  –1.20. The function has a local maximum at (0, 6) and a local minimum at (2, 2).
ANSWER

11. EXAMPLE 2
Find turning points
SOLUTION
b. g (x) 5 x4 – 6x3 + 3x2 + 10x – 3
a. Use a graphing calculator to graph the function.
Notice that the graph of g has four x-intercepts and three turning points.
You can use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.
The x-intercepts of the graph are x  –1.14, x  0.29, x  1.82, and x  The function has a local maximum at (1.11, 5.11) and local minimums at (–0.57, –6.51) and (3.96, – 43.04).
ANSWER

12. EXAMPLE 3
Maximize a polynomial model
You are making a rectangular box out of a 16-inch-by-20-inch piece of cardboard. The box will be formed by making the cuts shown in the diagram and folding up the sides. You want the box to have the greatest volume possible.
Arts And Crafts
• How long should you make the cuts?
• What is the maximum volume?
• What will the dimensions of the finished box be?

13. EXAMPLE 3
Maximize a polynomial model
SOLUTION
Write a verbal model for the volume. Then write a
function.

14. Maximize a polynomial model
EXAMPLE 3
Maximize a polynomial model
= (320 – 72x + 4x2)x
Multiply binomials.
= 4x3 – 72x x
Write in standard form.
To find the maximum volume, graph the volume
function on a graphing calculator. Consider only the interval 0 < x < 8 because this describes the physical restrictions on the size of the flaps.

15. EXAMPLE 3
Maximize a polynomial model
From the graph, you can see that the maximum
volume is about 420 and occurs when x  2.94.
You should make the cuts about 3 inches long.The
maximum volume is about 420 cubic inches. The dimensions of the box with this volume will be about x = 3 inches by x = 10 inches by x = 14 inches.
ANSWER

16. GUIDED PRACTICE
for Examples 1, 2 and 3
Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.
f (x) = x4 + 3x3 – x2 – 4x – 5
ANSWER
x-intercepts: –3.1, 1.4 local minimums: (–2.3, –9.6), (0.68, –7.0) local maximum: (–0.65, –3.5)

17. 5. WHAT IF? In Example 3, how do the answers
GUIDED PRACTICE
for Examples 1, 2 and 3
5. WHAT IF? In Example 3, how do the answers
change if the piece of cardboard is 10 inches by
15 inches?
(10  2x)
(15  2x)
= (150 – 50x + 4x2)x
Multiply binomials.
= 4x3 – 50x x
Write in standard form.
ANSWER
The cuts should be about 2 inches long. The maximum volume is about 132 cubic inches. The dimensions of the box would be 6 inches by 11 inches by 2 inches.

18. Daily Homework Quiz
2. You are making a rectangular box out of a 22- inch by 30-inch piece of cardboard, as shown in the diagram. You want the box to have the greatest possible volume. How long should you make the cuts? What is the maximum volume?
ANSWER
A square about 4.2 inches should be cut from each corner to produce a box with a maximum volume of about 1233 in.3