5.4Analyzing Graphs

of Polynomial Functions

Honors Algebra 2Mrs. Pam Miller

Graphing a Polynomial FunctionWe will NOT graph by hand….just on our calculators!

f(x) = -x4 + x3 + 3x2 + 2x

Is this an even or odd degree function?

Is the leading coefficient positive or negative?

How many REAL zeros does this function have?

How many imaginary solutions?

Describe the end behavior of the function.

NOW, graph the function on your calculator .

Location PrincipleAnalyzing sign changes to “locate” REAL zeros.

Look at the TABLE of ordered pairs that make up the graph of this polynomial function.

Look for sign changes in the “y” values.

Where do the REAL zeros occur?

You Try…Determine the consecutive integer values of x between which each real zero of this function is located.

f(x)= x4 – x3 – 4x2 + 1

Maximum & Minimum Points6

4

2

-2

-4

-6

-8

-10

-12

-14

5

RelativeMaximum

RelativeMinimum

This graph is the general shape of a 3rd degree polynomial function.

The maximum & minimum values of a function are called the EXTREMA.

These points are often referred to as TURNING POINTS.

The graph of a polynomial function of degree n has at most n – 1 turning points.

Find the Maximum & MinimumUse your calculator to find the relative maximum.

The relativemaximum is 5& it occurs atx = 0.

Find the Maximum & Minimum Use your calculator to find the relative minimum .

The relativeminimum is 1& it occurs atx = 2.

The graph of a polynomialfunction will have interval(s) over which it is increasing and interval(s) over which it is decreasing.

6

4

2

-2

-4

-6

-8

-10

-12

-14

-5 5

(2.90, -12.05)

(-.23, 3.24)

f(x) isdecreasing

f(x) isincreasing

f(x) isincreasing

RelativeMaximum

RelativeMinimum

Increasing Intervals:

Decreasing Interval:

Graph: 2x3 + x2 - 4x - 2 Answer these questions based on the graph.

1) Find the relative maximum & the x-coordinate atwhich it occurs.

2) Find the relative minimum & the x-coordinate atwhich it occurs.

3) State the Domain .

4) State the Range.

5) State the consecutive integer values of x betweenwhich each real zero is located.

6) Find the actual real zero(s) rounding to nearest hundredth as necessary.

7) State the interval(s) over which the function is increasing/decreasing.

4

2

-2

-4

-6

-5 5

(-1, 1)

(.67, -3.63)

All Real Numbers

All Real Numbers

Between : -2 & -1; -1 & 0; 1 & 2

-1.41, -.5, 1.41

Increasing: x < -1; x > .67Decreasing: -1< x < .67

Graph: f(x) = x4 – x3 – 4x2 + 1Answer these questions based on the graph.

1) Find the relative maximum & the x-coordinate at which it occurs.

2) Find the relative minima & the x-coordinate at which each occurs.

3) State the Domain .

4) State the Range.

5) How many REAL zeros exist? How many imaginary?

6) State the consecutive integer values of x between which each real zero is located.

7) Find the actual real zero(s) rounding to nearest hundredth as necessary.

8) Describe the end behaviors of the graph .

9) State the interval(s) over which the function is increasing/decreasing.

-1.05 at x = -1.09 & -7.31 at x = 1.84

1 at x = 0

All REAL Numbers

{y|y ≥ -7.31}

4 0

Between: -2 & -1; -1 & 0; 0 & 1; 2 & 3

- 1.44, -.57, .49, 2.52

x → + ∞, f(x) → + ∞; x → - ∞, f(x) → + ∞

Decreasing: x < -1.09; 0 < x < 1.84; Increasing: -1.09 < x < 0; x > 1.84