Name: Date:Advanced Math

Name____________________________________

Test Review: Rational Functions

Skills: Given a rational function, be able to find:

x – intercepts; y – intercept; coordinates of holes; horizontal/slant asymptotes (aka - end behavior asymptotes); vertical asymptotes

You should be able to sketch a graph of a rational function using the information found above and a few exact points.

You should be able to turn a function from the form f ( x )= ax+b

cx+d into the transformation

form f ( x )= a

x−h+k

. Then, identify the transformations from the parent function

g( x )=1x to f ( x ). Order is important - always put the vertical shift last.

You should also feel comfortable taking an equation in the form of f ( x )= a

x−h+k

and

returning it to a single rational expression in the form f ( x )= ax+b

cx+d Limits – be able to find limits at asymptotes, end behavior limits, and limits at any point.

Make sure to know how to use substitution and factoring as methods to evaluate limits.

Application: You should be able to match a graph to an equation. Use your knowledge of rational functions to make generalized statements, categorize

equations/graphs, create your own rational function given critical features Apply your knowledge to application problems. (Explain the meaning of asymptotes in the

context of the problem, find specific function values)

Notes to help you study:Finding end behavior asymptotes (horizontal asymptotes):

If degree is higher in the numerator, use long division to find the asymptote. If degree is higher in the denominator, there is a horizontal asymptote at y = 0. If degree is the same in the numerator and denominator, divide the coefficients of the

leading terms to find the horizontal asymptote.Finding x - intercepts: set numerator = 0 and solve for x.Finding y – intercept: plug in 0 for x.Finding vertical asymptotes: they are the zeros of the denominator unless they are a hole as described below:Finding holes:

Name: Date:Advanced Math

If factors in numerator and denominator cancel, there is a hole at the x value that makes that factor = 0. (to find the y – coordinate of that hole, plug x into the function AFTER you cancel out the factors)

PRACTICE:

1. Analyze and graph the functionf ( x )= ( x−3)

( x−3 )( x+4 ) . Include the x-intercepts, y – intercept, holes, and all asymptotes

2. Analyze each function below. Identify the x-intercepts, y – intercept, holes, and all asymptotes.

a. f ( x )= x

2+7 x−8x2+5

b. f ( x )=2x2+7x−4

2 x2+3 x−2

Name: Date:Advanced Math

c.f ( x )=x−2¿ x

2−1 ¿¿

¿

d. f ( x )= x

2+5 x−4x−2

3. Given f ( x )= 4 x+9

x+3

a. Express f(x) in transformation form. (Divide). (f ( x )= a

x−h+k

)

b. Identify the transformations to go from g( x )=1

x to f(x) in the proper order.

Name: Date:Advanced Math

c. Identify the domain

d. Write the equation for any vertical asymptotes.

e. Write the equation for any horizontal or slant asymptotes.

f. Find any x – intercepts. g. Find any y – intercepts.

4. The rabbit population of some farm is given by the function R( t )=3000t

t+1 , where t is in months since the start of the year.

a. When is population of rabbits equal to 2000?

b. What is limt→∞

R( t )equal to and what does it mean in the context of the problem?

c. What, if any, vertical asymptotes does this graph have, and what do they mean in the context of the problem?

5. A drug is injected into a patient’s bloodstream. The concentration of the drug t hours after

injection (in mg/liter) is given by the function c ( t )=30 t

t 2+2 .

a. What is limt→∞c ( t )

and what does it mean in the context of the problem?

Use your calculator to sketch a graph of c ( t )and answer the following:b. What is the maximum concentration and when does it occur?

Name: Date:Advanced Math

c. For what times is the concentration above 1 mg/liter?

d. When does the concentration drop below 0.5 mg/liter?

6. Use the graph of f ( x )below to find the following:

a. limx→∞

f ( x )b.

limx→4+

f ( x )c.

limx→4−

f ( x )d.

limx→10

f ( x )

e. f (10) f. limx→0f ( x )

g. f (0) h. limx→−2−

f ( x )

i. limx→−2+

f ( x )j.f (−2) k.

limx→−∞

f ( x )

Here are two graphs. Write everything you can tell about critical features down about each one. Then put that information into equation format. You will not have found the exact equation, but you will have some critical information that can help you determine what equation might be the right one in a matching situation.

7. VA

HA

x-int.

y – int.

Name: Date:Advanced Math

start of equation:

8. VA

HA

x-int.

y-int.

start of equation:

9. Write the equation of a rational function that has:

VA: x=3 and x = -4HA: y = 2X-int. (-1,0) and (x+6)y-int: (0,-1)