Graphing Polynomials

Graphing Polynomials

Advanced Algebra

Positive vs Negative

• Functions with a POSITIVE leading coefficient - RIGHT end goes UP

• Functions with a NEGATIVE leading coefficient - RIGHT end goes DOWN

Multiple Select

Which graphs have a POSITIVE leading coefficient? (Check all that apply)

Odd vs Even

• Functions with an ODD degree - ends go in OPPOSITE directions

• Functions with an EVEN degree - ends go in the SAME direction

Multiple Select

Which graphs have an ODD degree? (Check all that apply)

Multiple Choice

 $\$  Which could be the graph of  $f\left(x\right)=2x^3+3x^2+4\ ?$  

Multiple Choice

 $\$ Which could be the graph of  $f\left(x\right)=-3x^8+17x^6-14x^3+2x^2-1\ ?$   

Multiple Choice

 $\$  Which could be the graph of  $\ f\left(x\right)=-4x^5-19x^4+22x^2-x\ ?$  

Factors & Zeros

If in factored form a polynomial

f(x) = (x-a)(x-b)(x-c), then

(x-a), (x-b), and (x-c) are "factors" and

a, b, and c are zeros - they are the x-intercepts of the graph found at points (a,0), (b,0), and (c,0).

*The zeros are the opposite of what you see in the parenthesis!

Multiple Choice

 $\$  Which graph depicts the function  $f\left(x\right)=\left(x-6\right)\left(x-3\right)\left(x+8\right)?$  

Multiplicities (CBS)

MULTIPLCITY:          $\rightarrow$  ACTION: 
ONE                           $\rightarrow$  CUT through
Two (or any EVEN)   $\rightarrow$  BOUNCE
ODD (Any odd 3+)    $\rightarrow$   SQUIGGLE

Multiple Choice

 $\$  Which graph depicts the function  $f\left(x\right)=\left(x-2\right)^2\left(x-8\right)?$  

Multiple Choice

 $\$  Which graph depicts the equation  $f\left(x\right)\ =x^2\left(x+4\right)^3\left(x-3\right)?$  

Multiple Choice

Which is the equation for the graph shown?

•  $f\left(x\right)=x^3\left(x-5\right)^2$  

•  $f\left(x\right)=x^2\left(x-5\right)$  

•  $f\left(x\right)=x^3\left(x+5\right)^2$  

•  $f\left(x\right)=x^2\left(x+5\right)^2$  

End Behavior (Graphs)

The END BEHAVIOR of a function f describes the behavior of the graph of the function at the "ends" of the x-axis. 

 $x\rightarrow\infty$  as we look to the RIGHT
 $x\rightarrow-\infty$  as we look to the LEFT
 $f\left(x\right)\rightarrow\infty$  the graph goes UP
 $f\left(x\right)\rightarrow-\infty$  the graph goes DOWN

Multiple Choice

Express the END BEHAVIOR of the graph shown.

• 
  

  $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty\$

  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$
  

End Behavior (Equations)

 $x\rightarrow\infty$ , $f\left(x\right)\rightarrow$  +/- L.C.
 $x\rightarrow-\infty$ , $f\left(x\right)\rightarrow$ Even/Odd Deg

Multiple Choice

Express the END BEHAVIOR of the function:  $f\left(x\right)=-12x^7-19x^3+22$  

• 
  

  $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty\$

  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$
  

•  $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$ 
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$
  

Multiple Choice

Express the END BEHAVIOR of the function:  $f\left(x\right)=7x^4+9x^3-21$  

• 
  

  $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty\$

  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$
  

• $x\rightarrow\infty,\ f\left(x\right)\rightarrow\infty$
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$
  

•  $x\rightarrow\infty,\ f\left(x\right)\rightarrow-\infty$ 
  $x\rightarrow-\infty,\ f\left(x\right)\rightarrow\infty$