Rational Graph Characteristics

Rational Graph Characteristics

In this Lesson you will learn how to:

• Find x & y intercepts

• Find vertical & horizontal asymptotes

• How to find Domain and Range

X-Intercepts

• On a graph, this is where the graph touches/crosses the x-axis

• For an equation: Factor first, if able, then set each factor of numerator = 0

Multiple Choice

What is the x-intercept?

• (1,0)

• (-1,0)

• (0,1)

• (0,-1)

Multiple Choice

 $f\left(x\right)=\frac{x^2+2x-3}{x+5}$  What is/are the x-intercept(s) ?

(Hint: factor the numerator first, then set = 0)

•  $\left(0,-3\right),\ \left(0,\ 1\right)$  

•  $\left(-5,\ 0\right)$  

•  $\left(0,\ -5\right)$  

•  $\left(-3,\ 0\right),\ \left(1,\ 0\right)$  

Y-Intercepts

• Found the same way as other functions

• Plug in 0 for all x's and solve

Multiple Choice

 $f\left(x\right)=\frac{2x-3}{4x^2+2x+1}$   Find the y-intercept(s).

•  $\left(0,\ \frac{1}{2}\right)$  

•  $\left(\frac{1}{2},\ 0\right)$  

•  $\left(-3,\ 0\right)$  

•  $\left(0,\ -3\right)$  

Multiple Choice

What are the x and y intercepts?

• (4, 0) and (0, -4)

• (-4, 0) and (0, -4)

• (-4, 0) and (0, 4)

• (0, 0) and (0, -4)

Vertical Asymptotes: Graphs

• A Vertical Asymptote is an imaginary vertical line where the graph approaches but doesn't touch it

• In this example, it is  $x=-2$  

• Be careful! The dotted line is not always shown.  You may have to look for the imaginary 'wall'

Multiple Choice

What appears to be the vertical asymptote?

• x = -3

• y = -3

• x = -1

• y = -1

Multiple Select

There are 2 vertical asymptotes here. Which 2 are correct?

• x = -3

• x = -5

• x = -2

• x = 3

• x = 2

Vertical Asymptotes:  Equation

• The VA come from the undefined values (restricted values) that make the DENOMINATOR = 0

• Set DENOMINATOR = 0 (factor first) and solve

• Write the asymptote using:  $x=$  

Multiple Choice

What is the Vertical Asymptote?

• x= -5

• x= 5

• x= 6

• x= -6

Multiple Choice

What are the Vertical Asymptotes?

(Remember to FACTOR the denominator first!)

• x = 0, -5

• x=0, 5

• x = 2, -7

• x = -2, 7

Horizontal Asymptotes: Graphs

These are invisible horizontal lines that correspond to the Right/Left ENDS of the graphs. (For really small or really big values of x).

On this graph you can see the left and right sides of the graph seem to approach a y-value of 1.

So the HA is y = 1.

Multiple Choice

Let's Review....What appears to be the Vertical asymptote?

• y = -3

• x = 1

• y = -1

• y=3

Multiple Choice

Now, what appears to be the Horizontal asymptote?

• y = -3

• x = 1

• y = -1

• y=3

Multiple Choice

Which graph appears to have a Horizontal asymptote of y = -3?

Multiple Select

Which 2 graphs appear to have a Horizontal asymptote of y = 0?

Horizontal Asymptotes: Equation

• A Horizontal Asymptote can be found by comparing the DEGREE of the Numerator to the DEGREE of the Denominator

• There are 3 possibilities depending on which degree is higher/lower

How to find a Horizontal Asymptote

Horizontal Asymptotes always begin with  $y=$  

Multiple Choice

What is the Degree of the numerator and denominator?

• Num = 4 Denom = 1

• Num = 1 Denom = 1

• Num = 1 Denom = 2

• Num = 0 Denom = 1

Multiple Choice

The Degree of the top and bottom are the same. Simplify 4x/x to get the horizontal asymptote. What is it?

• y=4

• y=0

• y=-2

• y=1

Multiple Choice

What is the horizontal asymptote? (Note: the degree on the top and the bottom are the same)

• y = -4

• y = 1

• x = 1

• y = -6

Multiple Choice

Find the horizontal asymptote. (Note the Degree is bigger on top.)

• None

• y=-2

• y=2

• y=0

How to find Domain/Range:

• Domain- Read left to right and 'skip' over the Vertical Asymptote

• Range - Read bottom to top and 'skip' over the Horizontal Asymptote

Domain/Range of this graph?

• Domain- We 'skip' over the VA of -2, so the Domain is:  $\left(-\infty,\ 2\right)U\left(-2,\ \infty\right)$  

• Range - We 'skip' over the HA of 2, so the Range is:  $\left(-\infty,\ 2\right)U\left(2,\ \infty\right)$  

Multiple Choice

The vertical asymptote here x= -3, so what is the Domain?

•  $\left(-\infty,\ \infty\right)$ All Reals

•  $\left(-3,\infty\right)$ x-values bigger than -3

•  $\left(-\infty,3\right)U\left(3,\infty\right)$ All reals but x $\ne$  3

•  $\left(-\infty,-3\right)U\left(-3,\infty\right)$ All reals but x $\ne$   -3 
  

Multiple Choice

The horizontal asymptote here y=0, so what is the Range?

•  $\left(-\infty,\ \infty\right)$ All Reals

•  $\left(0,\infty\right)$ y-values bigger than 0

•  $\left(-\infty,0\right)U\left(0,\infty\right)$ All Reals but y $\ne$   0

•  $\left(-\infty,-3\right)U\left(-3,\infty\right)$ All reals but y $\ne$  -3 
  

Multiple Choice

What is the domain and range of this graph?

• D : All reals but x ≠ -3

  R: All reals but y≠ 1

• D : All reals but x ≠ 1

  R: All reals but y≠ -3

• D : All reals but x ≠ 3

  R: All reals but y≠ 1

• D : All reals but x ≠ -1

  R: All reals but y≠ 3

Great Job! You learned...

• How to find x & y intercepts

• How to find Vertical and Horizontal Asymptotes

• How to find Domain and Range

• Keep Practicing!!