1.2 Notes Part 2

1.2 Notes Part 2

Please follow along with this lesson and fill in the rest of your notes sheet.

End Behavior

What happens as the graph approaches either “end” of the x-axis
The value y approaches as x→±∞
*arrow means "approaches"

Limit notation:

 $\lim_{x\rightarrow\infty}f\left(x\right)=-\infty$   $\lim_{x\rightarrow-\infty}f\left(x\right)=-\infty$  

Try these on your own now finding the limit as x approaches infinity and negative infinity.

Answers

 $\lim_{x\rightarrow\infty}f\left(x\right)=\infty;\ \lim_{x\rightarrow-\infty}f\left(x\right)=-\infty$  

 $\lim_{x\rightarrow\infty}f\left(x\right)=-4;\ \lim_{x\rightarrow-\infty}f\left(x\right)=-4$  

Multiple Choice

Describe the end behavior of the graph.

• x →∞, f(x)→∞ and

  x→∞, f(x)→⁻∞

• x →∞, f(x)→∞ and

  x→⁻∞, f(x)→∞

• x →∞, f(x)→∞ and

  x→⁻∞, f(x)→0

• x →∞,f(x)→0 and

  x→∞, f(x)→⁻∞

Multiple Choice

In the above graph complete the following end behavior:
As x --> -∞, f(x) --> ____
As x --> +∞, f(x) --> ____

• -∞
  -∞
• +∞
  -∞
• -∞
  +∞
• +∞
  +∞

Symmetry- Even Function

xⁿ when n is even

f(-x)=f(x)

Symmetry with respect to y-axis

Symmetry- Odd Function

xⁿ when n is odd

f(-x)=-f(x)

Symmetry with respect to origin

Symmetry- Not a Function

Symmetry with respect to x-axis

Multiple Choice

Is this function even, odd or neither?

• Even

• Odd

• Neither

Multiple Choice

• Even Function

• Odd Function

• Function - Neither Even Nor Odd

• Both Even and Odd Function

• Not a Function

Multiple Choice

The function shown in the graph is:

• Even

• Odd

• Neither Even nor Odd

• Not a Function

Examples

Verify algebraically that  $f\left(x\right)=x^2-3$   is an even function.

*plug -x in for x and simplify

 $f\left(-x\right)=\left(-x\right)^2-3$  *squaring a negative is always positive, so
 $f\left(-x\right)=x^2-3$  
Which is exactly what we started with. f(-x)=f(x)

Try the next examples on your own

Answer

Neither odd nor even because f(-x) does not equal f(x) or -f(x)

Asymptotes

 $f\left(x\right)=\frac{1}{x+3};\ f\left(x\right)=\frac{x^2-9}{x+3}$  

Graph these two functions in the same screen of your graphing calculator or desmos.

Asymptotes

 $f\left(x\right)=\frac{1}{x+3};\ f\left(x\right)=\frac{x^2-9}{x+3}$  

Notice that -3 would make the denominatior of each of these functions 0, so -3 cannot be in the domian of either. 

Asymptotes

 $f\left(x\right)=\frac{1}{x+3};\ f\left(x\right)=\frac{x^2-9}{x+3}$  

The first function has an asymptotes at -3 and the second has a hole, which you cannot see on the graph.

Asymptotes

Vertical Asymptotes:
Occur in rational functions when the denominator equals 0.
x=0

Horizontal Asymptotes:
Find graphically for now
y=0

Try the Example on your own

Answer

VA: x=2, -2

HA: y=-2

Thank you for following along today! We will review this and look at more examples when we return to class.