Warm Up Oct. 16th

Solve each rational equation or inequality.

4 2

2

2x x 11) 0

x 9

x 12) 3

x 4

Homework Answers..1) x = 1, x = 3

2) 2 3

3 313)

2

4) x = 5

5) x = -2/3

7) , 5 U 2,1

7 382 9

8) ,

Rational Functions

Domain and Asymptotes

ObjectivesYou will be able to identify

– the domain– the y-intercept – the vertical asymptotes– the end behavior asymptote

of a rational function.

Let’s talk about the domain…

• Remember: “You can’t divide by zero”

• To find the domain of a rational function set the denominator equal to zero and solve for the real solutions. The domain is all real numbers except for those values.

Find the domain of each of the following rational functions.

1( )

2f x

x

2

12( )

5 6

xg x

x x

2

2 1( )

9

xh x

x

How do you find the y-intercept?

• Plug in x = 0 and simplify.

1( )

2f x

x

Find the y-intercept of each of the following rational functions.

2

12( )

5 6

xg x

x x

2

2 1( )

9

xh x

x

What about those vertical asymptotes?

• For a simplified rational function the vertical asymptote(s) are just the value(s) where the function is undefined. (look at the domain )

Write the equation for the vertical asymptote(s) of the following.

1( )

2f x

x

Write the equation for the vertical asymptote(s) of the

following.

2

12( )

5 6

xg x

x x

Write the equation for the vertical asymptote(s) of the following.

2

2 1( )

9

xh x

x

Wondering about the end behavior asymptotes?

• To find the end behavior asymptote simply divide the denominator into the numerator…the end behavior asymptote is the quotient.

But a “Hint” about those E.B.A.s• Bottom Heavy: degree of the bottom

is bigger than the degree of the top– EBA is always y = 0

• Equal:degree of the bottom is the same as the degree of the top– EBA is always y = leading coefficient

leading coefficient

• Top Heavy: degree of the top is bigger than the degree of the bottom– no trick you have to divide.

Identify the EBA for each of the following.

2 1( )

3

xf x

x

2

12( )

5 6

xg x

x x

2 3 5( )

2

x xh x

x

What are the equations of the asymptotes?

Describe the equation of the graph.

What are the equations of the asymptotes?

Describe the equation of the graph.

What are the equations of the asymptotes?

Describe the equation of the graph.

Find your partner…

Match each graph to the correct equation…

The finance committee of a nonprofit summer camp for children is setting the cost for a 5-day camp. The fixed cost for the camp is $2400 per day, and includes things such as rent, salaries, insurance, and equipment.

An outside food services company will provide meals at a cost of $3 per camper, per meal. Campers will eat 3 meals a day.

As a nonprofit camp, the camp must cover its costs, but not make any profit.

The committee must come up with a proposal for setting the fee for each camper, based on the number of campers who are expected to attend each week.

Initially, the committee decides to calculate camper fees based on the fixed cost of the camp alone, without meals for the campers.

What is the total fixed cost for the five days?

Write a function for the fee per camper(without food) as a function of the number of campers in attendance.

The function developed in Item 1 did not account for meals. Campers eat three meals per day at a cost of $3 per camper per meal. The committee must determine a function that includes the cost of mealswhen setting the fee per camper.

What will be the total cost for meals per camper each week?

Write a function for the fee per camper(with food) as a function of the number of campers in attendance.

The committee decides to award 30 scholarships to students who otherwise could not afford the camp. These scholarships include full use of the facilities and all meals at no charge.

Write a function for the fee per paying camper(with food) as a function of the number of campers in attendance.

The camp can accommodate up to 300 campers, and market research indicates that campers do not want to pay more than $200 per week. Although the camp is nonprofit, it cannot afford to lose money.

Write a proposal for setting the fee per camper. Be sure to include these items.• the proposed fee• the minimum number of campers needed to break even• the maximum possible income for the proposed fee• mathematics to support your reasoning