6.8 Analyzing Graphs of Polynomial Functions

Analyzing Graphs of Polynomial Functions

The relationships between zeros, factors, solutions, and x-intercepts are summarized below.

Let f (x) = a n x n + a n – 1 x n – 1 + . . . + a 1x + a 0 be a polynomial function.

The following statements are equivalent.

ZERO: k is a zero of the polynomial function f.

FACTOR: x – k is a factor of the polynomial f (x).

SOLUTION: k is a solution of the polynomial equation f (x) = 0.

X-INTERCEPT: k is an x-intercept of the graph of the polynomial function f.

ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTSCONCEPT

SUMMARY

If k is a real number, then the following is also equivalent.

Using x-Intercepts to Graph a Polynomial Function

Graph the function f (x) = 14

(x + 2)(x – 1) 2.

Plot x-intercepts. Since x + 2 and x – 1 are factors of f (x), –2 and 1 are the x-intercepts of the graph of f. Plot the points (–2, 0) and (1, 0).

511–4

320–1–3–4

y

x

12

12

–12

SOLUTION

14

Because f (x) has three linear factors of the form x – k and a constant factor

of , it is a cubic function with a positive leading coefficient.

Plot points between and beyond the x-intercepts.

Determine the end behavior of the graph.

Using x-intercepts to Graph a Polynomial Function

Draw the graph so it passes through the points you plotted and has the appropriate end behavior.

Therefore, f(x) – – as x

f(x) as x and

Graph the function14

(x + 2)(x – 1) 2.

Remember End-Behavior

• Positive L.C, Even Degree (both up)• Positive L.C, Odd Degree (Right up, Left down)• Negative L.C., Even Degree (both point down)• Negative L.C., Odd Degree (Left up, Right down)

UP – UP (POSITIVE)DOWN – DOWN (NEGATIVE)

EVEN DEGREES:

ODD DEGREES:RISE TO THE RIGHT (POSITIVE)FALLS TO THE RIGHT (NEGATIVE)

Graph the Functionf x x x( ) ( )( ) 1 3

Plot the intercepts.

Make a table and plot some points.

What are the end behaviors?

What kind of function is it?

Graph the Functionf x x x x( ) ( )( )( ) 3 1 1

Plot your zeros. Multiply the factors together to tell you about end behavior

Analyzing Polynomial Graphs

TURNING POINTS Another important characteristic of graphs of polynomial functions is that they have turning points corresponding to local maximum and minimum values.

The y-coordinate of a turning point is a local maximum of the function if the point is higher than all nearby points.

The y-coordinate of a turning point is a local minimum of the function if the point is lower than all nearby points.

TURNING POINTS OF POLYNOMIAL FUNCTIONS

The graph of every polynomial function of degree n has at most n – 1 turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly n – 1 turning points.

Estimate the coordinates of the turning point. Is it a local

minimum or a local maximum?

What degree is the function?

Click on the link…

•Polynomial Function Graphs

•Use the website to determine what the coefficients of the terms do.

Day 2:

Using Polynomial Functions in Real Life

You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by making the cuts shown in the diagram and folding up the sides so that the flaps are square. You want the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be?

Using Polynomial Functions in Real Life

You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by making the cuts shown in the diagram and folding up the sides so that the flaps are square. You want the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be?

Labels

…

Volume = V

Width = 10 – 2 x

Length = 15 – 2 x

Height = x

(cubic inches)

(inches)

(inches)

(inches)

SOLUTION

Verbal Model

Volume = Width Length Height

Algebraic Model

…

Using Polynomial Functions in Real Life

You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by making the cuts shown in the diagram and folding up the sides so that the flaps are square. You want the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be?

V = (10 – 2 x) (15 – 2 x) x

= (4 x 2 – 50 x + 150) x

= 4 x 3 – 50 x 2 + 150 x

Using Polynomial Functions in Real Life

You are designing an open box to be made of a piece of cardboard that is 10 inches by 15 inches. The box will be formed by making the cuts shown in the diagram and folding up the sides so that the flaps are square. You want the box to have the greatest volume possible. How long should you make the cuts? What is the maximum volume? What will the dimensions of the finished box be?

To find the maximum volume, graph the volume function. Consider only the interval 0 < x < 5 because this describes the physical restraints on the size of the flaps. From the graph, you can see that the maximum volume is about 132 and occurs when x 1.96.

You should make the cuts approximately 2 inches long. The maximum volume is about 132 cubic inches.

The dimensions of the box with this volume will be x = 2 inches by 10 – 2x = 6 inches by 5 – 2x = 11 inches.