Domains and Graphs of Rational Functions: Lesson

30

LESSON OBJECTIVES:

1. Create rational functions to model situations.

2. Analyze graphs of rational functions and their asymptotes.

1) Direct Variation: A linear function in the form y = kx, where k 0.

2) Constant: A linear function in the form y = b.

3) Identity: A linear function in the form y = x.

4) Absolute Value: A function in the form

y = |mx + b| + c (m 0).

5) Greatest Integer: A function in the form y = [x].

6) Rational Function: A function that is the ratio of two polynomials. The polynomial you are dividing by cannot be zero. A function in the form

y = (x + 3)(x – 5) (x – 3)

7) Square Root Function: function that maps the set of non-negative real numbers onto itself. It is continuous for all non-negative x and differentiable for all positive x. It is in the form y = .

8) Asymptote: A line that a graph of a function approaches but never crosses or intersects.

9) Inverse Function: a function obtained by expressing the dependent variable of one function as the independent variable of another; f and g are inverse functions if f(x)=y and g(y)=x. Function in the form of y = .

10)Point Continuity: Hole in a graph.

11)Continuity: Can be traced with a pencil never leaving the paper.

12)Vertical Asymptotes: x = 3. Function grows to infinity.

13)Horizontal Asymptotes: y = 8. Function approaches as x tends to plus or minus infinity.

Direct Variation: A linear function in the form y = kx, where k 0.

Constant: A linear function in the form y = b.

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

y = 3

Identity: A linear function in the form y = x.

Absolute Value: A function in the form y = |mx + b| + c (m 0).

Absolute Value: A function in the form y = |mx + b| + c (m 0).

Greatest Integer: A function in the form y = [x].

Rational Function: A function that is the ratio of two polynomials.

Square Root Function: Function that is in the form of y = .

Inverse Function: Function in the form of y = .

Vertical and Horizontal Asymptotes

Suppose concert promoters want to increase the profit per ticket for various possible ticket prices.

Number of tickets sold:

Profit from ticket sales:

1) What number of tickets, total profit, and profit per ticket would be expected:

a. If the average ticket price is $10.

b. If the average ticket price 𝑥is $20.

2) Sketch graphs of the equations and answer the following questions.

a. What is the practical domain of ? That is, what prices will give predicted numbers of tickets that make sense in the concert situation?

b. Estimate the ticket price for which profit from concert operation P ( ), excluding 𝑐 𝑥snack bar operations, is maximized. Find that profit.

3) Now turn to analysis of the profit-per-ticket function .

a. Graph for between 0 and 30.

b. Estimate the ticket price for which profit per ticket is maximized.

c. Why is the profit per ticket not maximized at the same ticket price that maximizes concert profit?

Study the graph handout of the rational function and describe the pattern of change in function values as approaches 3.𝑥The graph has = -1 and = 3 𝑥 𝑥as vertical asymptotes. As values of approach -1 from below, the 𝑥graph suggests that values of f(𝑥) decrease without lower bound.

As values of approach -1 from 𝑥above, the graph suggests that values of f( ) increase without 𝑥upper bound.

As values of approach 3 from 𝑥below 3, the function values decrease rapidly approaching negative infinity. As values of 𝑥approach 3 from above 3, the function values approach positive infinity.

HOMEWORK!!1) “Graphing Rational

Functions” Worksheet.2) “Domains and Graphs of

Rational Functions” Worksheet.