5.5 Analyze Graphs of Polynomial Functions

5.5 Analyze Graphs of Polynomial Functions

Cubic Function (3rd degree Polynomial)

The degree of the polynomial tells you how many times the curve changes direction.

 $y=2x^3-2x^2-4x$  

Find x-intercepts by Factoring

Multiple Choice

What is the completely factored form?

 $2x^3-2x^2-4x$  

•  $2x\left(x-2\right)\left(x+1\right)$  

•  $2x\left(x^2-x-2\right)$  

•  $x^2-x-2$  

•  $\left(x-2\right)\left(x+1\right)$  

4th Degree Polynomial

 $y=x^4+2x^3-x^2-2x$  

Factor by Grouping

Multiple Choice

 $x^4+2x^3-x^2-2x$  

What is the completely factored form?

•  $\left(x^3-x\right)\left(x+2\right)$  

•  $x\left(x^2-1\right)\left(x+2\right)$  

•  $x\left(x+1\right)\left(x-1\right)\left(x+2\right)$  

•  $x\left(x+1\right)\left(x-1\right)\left(x-2\right)$  

Multiple Choice

What is the degree of the polynomial function?

• 2

• 3

• 4

• 5

Multiple Choice

Which equation could be for the function in the diagram?

• 
  

  $y=x^3-2x^2$
  

• $y=x^4+2x^3-3x^2$

• $y=x^5$

• $y=x^2+2x$

Multiple Choice

Which equation could be for the function in the diagram?

• $y=x^3+2x^2+1$

• $y=x^4+x^3-2x^2$

• $y=x^5-2x^4$

• $y=3x^3+4$

X-intercepts

You can write an equation for a polynomial function by identifying the x-intercepts, and writing the equation as a product of factors.

 $y=\left(x-1\right)\left(x+1\right)\left(x-2\right)$  

Write an equation for the graph by writing the x-intercepts as factors

 $y=x\left(x-1\right)\left(x-2\right)\left(x-3\right)$  

Multiple Choice

Which equation matches the graph?

• 
  

  $y=\left(x+1\right)\left(x+2\right)$
  

• $y=\left(x+1\right)\left(x-2\right)$

• $y=x\left(x+2\right)$

• $y=\left(x+2\right)\left(x-1\right)$

• $y=x^2+3x+2$

Multiple Choice

Which equation matches the graph?

• $y=\left(x-3\right)\left(x+2\right)\left(x-1\right)$

• $y=\left(x-3\right)\left(x-2\right)\left(x-1\right)$

• $y=\left(x+3\right)\left(x-2\right)\left(x+1\right)$

• $y=x\left(x-1\right)\left(x+2\right)\left(x+3\right)$

Summary

• The Degree of a Polynomial tells us how many times the graph changes direction, and the maximum number of x-intercepts.

• You can write x-intercepts as factors (i.e. if the graph has x-intercept at 5, this means the factor is (x-5).

• In order to find x-intercepts, you must factor the polynomial, set it equal to zero, and solve for x.

• Since most polynomials are not factorable, graphing calculators have features which allow you to find x-intercepts.