Relative (Local) Max and Min of a Polynomial Real Zeros of a ...

NAME: __________________________________________________ DATE: _________________

Algebra 2: Lesson 7-2 Sketching a Polynomial Functions

Learning Goals:

1) How do we sketch a polynomial by using its characteristics?

Relative (Local) Max and Min of a Polynomial

Real Zeros of a Polynomial The zeros of a polynomial function are the solutions to the polynomial equation

when the polynomial equals zero

The real zeros of a polynomial are where the polynomial graph crosses the x-axis.

Y-Intercept of a Function We have talked about -intercepts of the graph of a function in previous lessons. The -intercepts

correspond to the zeros of the function.

Consider the following examples of polynomial functions and their graphs to determine an easy way to

find the -intercept of the graph of a polynomial function.

( ) ( ) ( )

Directions: Without using a graphing utility, match each graph below in column 1 with the function in

column 2 that it represents.

a.

1.

b.

2.

c.

3.

d.

4.

e.

5.

Highest Degree of Polynomial and End Behavior

1. Sketch the graph of ( ) . What will the graph of ( ) look like? Sketch it on the same

coordinate plane. What will the graph of ( ) look like?

2. Sketch the graph of ( ) . What will the graph of ( ) look like? Sketch this on the same

coordinate plane. What will the graph of ( ) look like? Sketch this on the same coordinate plane.

3. Consider the following function, ( ) , with a mixture of odd and even

degree terms. Predict whether its end behavior will be like the functions in Example 1 or Example 2.

Graph the function using a graphing utility to check your prediction.

( ) ( )

Examples: For each graph, determine whether it represents an odd-degree or an even-degree

polynomial and determine the sign of the leading coefficient (positive or negative).

Putting It All Together (Sketching a Graph) To graph a polynomial function, first plot points to determine the shape of the graph’s middle portion

by using the characteristics given (zeros, intervals of increase/decrease, max/min, y-intercept).

Then connect the points with a smooth continuous curve and use what you know about the end

behavior to sketch the graph.

Summary of Characteristics of a Polynomial Graph 1. Real Zeros – these are the x-intercepts of the graph

2. Y-intercept – this is where the graph crosses the y-axis (when x = 0)

3. Relative (Local) Max/Min – these are the turning points of the graph. They help to determine

intervals for which a function is increasing or decreasing.

4. End Behavior – describes the behavior of a graph as approaches . For a graph of a

polynomial function, the end behavior is determined by the function’s degree and the sign of its

leading coefficient.

Example: Sketch a graph of a polynomial function having these characteristics.

is increasing when and .

is decreasing when .

( ) when and .

( ) when and .

Directions: Sketch a graph of a polynomial function having these characteristics.

1. It is an even-degree polynomial function with a negative leading coefficient.

The graph intersects the x-axis at two points.

2. ( ) . ( )

It is an odd-degree polynomial function.

The graph intersects the x-axis once.

3. The graph of has x-intercepts at .

has a local maximum value when .

has a local minimum value when .

4. is increasing when .

is decreasing when

( ) when and .

( ) when .

NAME: _______________________________________________ DATE: __________________

Algebra 2: Homework 7-2

1. Sketch a graph of a polynomial function having these characteristics:

a) An even-degree polynomial with one relative maximum and two relative minimums

b) The graph of has x-intercepts at . has a local maximum value when . has a local minimum value when and .

c) is increasing when . is decreasing when and . ( ) when and . ( ) when and .

2. Describe the end behavior of the graph of the function.

a) ( ) b) ( )

c) ( ) d) ( )

3. For each graph, determine whether it represents an odd-degree or an even-degree polynomial and

determine the sign of the leading coefficient (positive or negative).

4. Consider the partial graph of the function ( ) shown twice below.

Sketch the other half of the function if in (a) ( ) is even and in (b) ( ) is odd.

5. If ( ) is an odd, one-to-one function and if ( ) , then which of the following points must lie on the

graph of the inverse of ( ), ( )?

(1) (-7, 2) (3) (2, 7)

(2) (2, -7) (4) (7, -2)

6. Algebraically, determine whether the following function is odd, even, or neither.

( )

Midterm Review

7. Express the following in simplest a + bi form: ( ) ( )

8. Factor completely:

9. Express ( ) in simplest form with positive exponents only.