Graphing Polynomial Functions

Goal:

Evaluate and graph polynomial functions.

CCSS: F.IF.4

Given a function, identify key features in graphs and tables including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the

reasoning of others.   4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Essential Question:

How do I graph a polynomial function? 

Polynomial Function

Function of the form:

an is the leading coefficient a0 is the constant term n is the degree Polynomial is in standard form if its terms are written in

descending order of exponents from left to right.

11 1 0( ) .........n n

n nf x a x a x a x a

EVALUATING POLYNOMIAL FUNCTIONS

A polynomial function is a function of the form

f (x) = an x n + an – 1 x

n – 1 +· · ·+ a 1 x + a 0

Where an 0 and the exponents are all whole numbers.

A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

For this polynomial function, an is the leading coefficient,

a 0 is the constant term, and n is the degree.

an 0

an

an leading coefficient

a 0

a0 constant term n

n

degree

descending order of exponents from left to right.

n n – 1

Common Types of Polynomials

Degree Type Standard Form

0 Constant f(x) = a0

1 Linear f(x) = a1x + a0

2 Quadratic f(x) = a2x2 + a1x + a0

3 Cubic f(x) = a3x3 + a2x2 + a1x + a0

4 Quartic f(x) = a4x4 + a3x3 + a2x2 + a1x + a0

Identifying Polynomial Functions

2 4

3

1. ( ) 3 7

2

. ( ) 3x

a f x x x

b f x x

Decide whether the function is a polynomial function. If it is, write the function inStandard form and state its degree, type, and leading coefficient.

Identifying Polynomial Functions

2 1

2

. ( ) 6 2

. ( ) 0.5 2

c f x x x x

d f x x x

Decide whether the function is a polynomial function. If it is, write the function inStandard form and state its degree, type, and leading coefficient.

Evaluate the Polynomial FunctionUsing Synthetic Substitution

3 2( ) 5 4 8 1 when x = 2f x x x x

Evaluating a Polynomial Function in Real Life

The time t (in seconds) it takes a camera battery to recharge after flashing n times can be modeled by:

Find the recharge time after 100 flashes.

3 2( ) 0.000015 0.0034 0.25 5.3f x n n n

6.2 Continued: Graphing Polynomial Functions

Will use end behavior to analyze the graphs of polynomial functions.

End Behavior

Behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞)

The expression x→+∞ : as x approaches positive infinity

The expression x→-∞ : as x approaches negative infinity

End Behavior of Graphs of Linear Equations

f(x)→+∞ as x→+∞ f(x)→-∞ as x→-∞

f(x) = x

f(x)→-∞ as x→+∞ f(x)→+∞ as x→-∞

f(x) = -x

End Behavior of Graphs of Quadratic Equations

f(x)→+∞ as x→+∞ f(x)→+∞ as x→-∞

f(x) = x²

f(x)→-∞ as x→+∞ f(x)→-∞ as x→-∞

f(x) = -x²

Investigating Graphs of Polynomial Functions

1. Use a Graphing Calculator to grph each function then analyze the functions end behavior by filling in this statement: f(x)→__∞ as x→+∞ and f(x)→__∞ as x→-∞

a. f(x) = x³ c. f(x) = x4 e. f(x) = x5 g. f(x) = x6

b. f(x) = -x³ d. f(x) = -x4 f. f(x) = -x5 h. f(x) = -x6

Investigating Graphs of Polynomial Functions

How does the sign of the leading coefficient affect the behavior of the polynomial function graph as x→+∞?

How is the behavior of a polynomial functions graph as x→+∞ related to its behavior as x→-∞ when the functions degree is odd? When it is even?

End Behavior for Polynomial Functions

For the graph of

If an>0 and n even, then f(x)→+∞ as x→+∞ and f(x)→+∞ as x→-∞

If an>0 and n odd, then f(x)→+∞ as x→+∞ and f(x)→-∞ as x→-∞

If an<0 and n even, then f(x)→-∞ as x→+∞ and f(x)→-∞ as x→-∞

If an<0 and n odd, then f(x)→-∞ as x→+∞ and f(x)→+∞ as x→-∞

11 1 0( ) .........n n

n nf x a x a x a x a

Graphing Polynomial Functions

f(x)= x³ + x² – 4x – 1

x -3 -2 -1 0 1 2 3

f(x)

Graphing Polynomial Functions

f(x)= -x4 – 2x³ + 2x² + 4x

x -3 -2 -1 0 1 2 3

f(x)