1. Solving and Graphing Rational Functions

2. Rational Equations A rational equation is an equation that has one
or more rational expressions.
Examples:

3. One way is to solve is to Cross Multiply!

4. Example

5. Example

6. Example

7. Solve by Graphing 1. Enter right hand side of equation into Y1 and left hand side of equation into Y2 2. Graph on an appropriate viewing window 3. Determine point(s) of intersection 4. Convert decimals to fraction solutions if needed

8. Example

9. Example

10. Rational Function
A function in the form: 𝑅 𝑥 = 𝑝(𝑥) 𝑞(𝑥)
The functions p and q are polynomials.
The domain of a rational function is the set of all real numbers except those values that make the denominator, q(x), equal to zero.
𝑓 𝑥 = 2 𝑥 2 −4 𝑥+5
ℎ 𝑥 = 2 𝑥 2 −4
𝑔 𝑥 = 𝑥 2 −1 𝑥−1

11. Domain of a Rational Function
𝑓 𝑥 = 𝑥 2 −3 𝑥+4
𝑥+4=0
𝑥=−4
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –4} or
(-, -4)  (-4, )

12. Domain of a Rational Function
𝑔 𝑥 = 𝑥 2 −4 𝑥−2
𝑥−2=0
𝑥=2
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  2} or
(-, 2)  (2, )

13. Domain of a Rational Function
ℎ 𝑥 = 2 𝑥 2 −9
𝑥 2 −9=0
(𝑥−3)(𝑥+3)=0
x=−3, 3
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –3, 3} or
(-, -3)  (-3, 3)  (3, )

14. Domain of a Rational Function
ℎ 𝑥 = 2𝑥+3 𝑥 2 −2𝑥−15
𝑥 2 −2𝑥−15=0
(𝑥−5)(𝑥+3)=0
x=−3, 5
𝐷𝑜𝑚𝑎𝑖𝑛:
{x | x  –3, 5} or
(-, -3)  (-3, 5)  (5, )

15. Linear Asymptotes
Lines in which a graph of a function will approach.
Vertical Asymptote
A vertical asymptote exists for any value of x that makes the denominator zero AND is not a value that makes the numerator zero.
Example
𝑓 𝑥 = 𝑥 2 −16 𝑥+5
= (𝑥−4)(𝑥+4) 𝑥+5
x=−5
A vertical asymptotes exists at x = -5.
VA: 𝑥=−5

16. Asymptotes
Vertical Asymptote
Example
𝑓 𝑥 = 𝑥 2 −𝑥−6 𝑥 2 −7𝑥+12
= (𝑥+2)(𝑥−3) (𝑥−4)(𝑥−3)
x=3, 4
A vertical asymptote exists at x = VA: 𝑥=4
A vertical asymptote does not exist at x = 3 as it is a value that also makes the numerator zero.
A hole exists in the graph at x = 3.

17. Asymptotes
Horizontal Asymptote
A horizontal asymptote exists if the largest exponents in the numerator and the denominator are equal, or
if the largest exponent in the denominator is larger than the largest exponent in the numerator.
If the largest exponent in the denominator is equal to the largest exponent in the numerator, then the horizontal asymptote is equal to the ratio of the coefficients.
If the largest exponent in the denominator is larger than the largest exponent in the numerator, then the horizontal asymptote is 𝑦=0.

18. Asymptotes
Horizontal Asymptote
Example
𝑓 𝑥 = 5𝑥 3 −2 𝑥 2 −7 2𝑥 3 −7𝑥+10
HA: 𝑦= 5 2
A horizontal asymptote exists at y = 5/2.
𝑓 𝑥 = 𝑥−6 𝑥 2 −7𝑥+12
A horizontal asymptote exists at y = 0.
HA: 𝑦=0

19. Holes
When a value of x sets both the numerator and the denominator equal to zero, then a hole is created. This means that there is a single point on the function that is not covered by the domain.

20. Holes Example The following functions have holes in their graphs:
f (x) = 
                       
f (x) =               

21. Name all asymptotes and holes: