Name: Rational Functions 2.1H Readymsshultis.weebly.com/uploads/1/0/9/3/10930910/fall...1. 2. Yes Yes ( ) 3. 4. No Yes ( ) Set Topic: Simplifying rational expressions & operations

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SDUHSD Math 3 Honors

Name: Rational Functions 2.1H

Ready, Set, Go!

Ready Topic: Polynomial division Use division to determine if the given linear term is a factor of the polynomial. If it is a linear factor, then find the other factors of the polynomial. 1. 2. Yes Yes

( )

3. 4. No Yes ( )

Set Topic: Simplifying rational expressions & operations on rational expressions Answer the following questions. 5. Define rational expression and give three examples. A rational expression is an expression that can be written as the quotient of two polynomials. 6. Circle the expressions below that are rational. Explain why the non-circled terms are irrational.

√

isn’t rational because is not a polynomial. √ isn’t rational because it isn’t a

polynomial.

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7. Angela simplified the following rational expressions. Circle the one(s) she answered correctly. Then, identify and correct where she went wrong in the other two problems.

a.

b.

c.

A: Distributed incorrectly in the 3rd line. Answer should be

.

B: Canceled the terms in the 2nd line when she needed a common denominator. Answer

should be:

.

Simplify each expression. Leave answers in factored form.

8.

9.

10.

11.

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Topic: Solving rational equations Find ALL solutions to the following equations. Watch out for extraneous solutions (answers that make the original equation false).

12.

13.

is extraneous

14.

15.

no solution

is extraneous

16.

17.

no solution

is extraneous Topic: Solving rational inequalities 18. Which of the following is/are true?

a. The solution set of is .

b. The inequality

can be solved by multiplying both sides by , resulting in the equivalent

inequality .

c. and

have the same solution set.

d.

has no real solution.

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Solve each rational inequality. Write your answers in interval notation.

19.

20.

]

21.

22.

] ]

Go Topic: Solving radical equations Find ALL solutions to the following equations. Watch out for extraneous solutions (answers that make the original equation false).

23. √ 24. √

is extraneous

25. √ 26. √ is extraneous

27. a. What causes extraneous solutions when solving radical equations?

If the solution makes the radicand negative. b. What causes extraneous solutions when solving rational equations?

If the solution causes the denominator to equal 0.

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Topic: Trigonometric ratios in a right triangle Complete the given trigonometric ratios for .

28.

29.

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Ready, Set, Go!

Ready Topic: Operations on rational expressions Simplify each of the following expressions.

1.

2.

3.

4.

5.

6.

Topic: Proper vs. improper fractions Classify each fraction as proper or improper. If the fraction is improper, rewrite as a mixed number.

7.

improper,

8.

proper

9.

improper,

10.

improper,

11.

improper,

12.

proper

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Set Topic: Features and families of functions Find the roots and domain of each function. State the equations of any vertical asymptotes, if they exist. 13. 14.

Zeros: Zeros:

Domain: Domain: Asymptotes: None Asymptotes: None

15.

16.

Zeros: Zeros: Domain: Domain: Asymptotes: Asymptotes:

17. √ 18.

Zeros: Zeros:

Domain: Domain: Asymptotes: None Asymptotes: None

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Topic: Features of rational functions For each function, fill in the table of values and then graph the function. Then list the features of the function (domain/range, continuous/not continuous, intercepts, etc.).

19.

0 Und

2

1 1

2

List of Features: Domain: Range: Continuous: not continuous Intercepts: none Asymptotes: End behavior: As As

20.

Und

2

1

0

List of Features: Domain: Range: Continuous: not continuous Intercepts: no x-intercepts

y-intercept: (

)

Asymptotes: End behavior: As As

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21.

0 Und

6

1 3

2


22.

1

2

0 Und

1


23. What happens to as the x-values get closer to zero in the graph in question 22? The values of approach

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Go Topic: Solving inequalities Solve each inequality by placing the zeros of the related equation on a number line and checking a value in each interval. Express your solutions to the inequality in interval notation.

Example:

Related equation: Zeros of the related equation:

Solution to the inequality: [ ] [

24. 25.

*

+

26.

27.

]

28.

(

)

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Ready, Set, Go!

Ready Topic: Distinguishing between proper and improper rational functions. Determine if each of the following is a proper or an improper rational function. (Hint: look at the degree of the polynomials.)

1.

2. 3.

Improper Improper Proper

4.

5.

Improper Proper 6. Which of the above functions have the following end behavior:

as and as

Questions 3 & 5 7. Complete the statement: ALL proper rational functions have end behavior that… approaches zero Topic: Describing end behavior of polynomial functions Based on the equations alone, describe what happens as and . 8. 9. 10. as as as as as as

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Set Topic: Features of rational functions. Find the x-intercept(s), y-intercept, and any vertical asymptotes of the following functions.

11.

12.

( )

x-intercept(s): x-intercept(s):

y-intercept:

y-intercept:

vertical asymptotes: vertical asymptotes:

Topic: Improper vs. proper rational expressions Determine if each rational expression is proper or improper. If improper, divide the polynomials to

rewrite the rational expressions such that

where represents the quotient and

represents the remainder.

13.

14.

Improper Proper

15.

16.

Improper Improper

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Go Topic: Simplifying rational expressions Simplify each of the rational expressions by canceling common factors. Leave answers in factored form, where possible.

17.

18.

19.

20.

21.

( )

Topic: Solving rational equations and inequalities Solve each rational equation. Be sure to check for extraneous solutions.

22.

23.

No solution Extraneous: Extraneous:

Solve each rational inequality. Write your answers in interval notation.

24.

25.

[ [ ]

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Ready, Set, Go!

Ready Topic: Domain and range Based on the graph given in each problem below, identify the domain and range of each function. 1.

Domain: [ Range: ]

2.

Domain: ] Range:

3.

Domain: Range:

4.

Domain: [ ] Range: [ ]

Set Topic: Determine end behavior for rational functions For each of the given functions, describe the end behavior as the x-values approach and also .

5.

6.

As As As As

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7.

8.

As As As As Topic: Finding vertical and horizontal asymptotes of rational functions. Find the vertical and any horizontal asymptotes for the functions below.

9.

10.

Vertical: Vertical: Horizontal: Horizontal:

11.

12.

Vertical: Vertical:

Horizontal: Horizontal:

13.

14.

Vertical: Vertical:

Horizontal:

Horizontal: none

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Topic: Even and odd functions 15. Determine which of the following functions are even, odd or neither. Label them accordingly.

a. b.

c.

even odd neither d. | | e. f. even even neither 16. Use technology to graph each of the functions from number 4. What graphical characteristics go with an

even function and what characteristics go with an odd function?

Even functions are symmetric about the y-axis. Odd functions have 180 rotational symmetry. Neither functions do not have the symmetries listed.

Given that the partial graphs below are even and odd functions, draw in the rest of the graph. 17. Even function 18. Odd Function

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Go Topic: Rational equations and inequalities. Solve each rational equation and inequality. 19.

20.

21.

22.

( )

23.

√ (

)

Extraneous: Topic: Simplifying rational expressions Simplify each expression. Where possible, leave your answers in factored form.

24.

25.

26.

27.

( )

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Ready, Set, Go!

Ready Topic: Solving right triangles Find the indicated values for the geometric figures below.

1. Solve the right triangle. ̅̅ ̅̅ ̅̅ ̅̅

2. Solve the right triangle. ̅̅ ̅̅

Topic: Perimeter and area of regular polygons Find the perimeter and area of each regular polygon.

3. Perimeter: Area: 110.1106

4. Perimeter: Area:

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Set Topic: Features of rational functions Identify the features of the function, complete the sign line, and then sketch the function.

5.

6.

Domain: Domain:

Intercepts: x-int: 0, y-int: 0 Intercepts: y-int:

End Behavior: as End Behavior: as as as Vertical Asymptote(s): Vertical Asymptote(s): Horizontal Asymptote: Horizontal Asymptote: Sign Line: Sign Line:

Graph: Graph:

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7.

8.

Domain: Domain: Intercepts: x-int: 2, y-int: 2 Intercepts: None End Behavior: as End Behavior: as as as Vertical Asymptote(s): Vertical Asymptote(s): Horizontal Asymptote: Horizontal Asymptote: Sign Line: Sign Line:

Graph: Graph:

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Go Topic: Simplifying rational expressions Rewrite each of the rational expressions in its simplest form. Where possible, leave answers in factored form.

9.

10.

11.

( )

12.

13.

14.

Topic: Solving rational equations Solve each equation.

15.

16.

√ is extraneous

17.

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Ready, Set, Go!

Ready Topic: Solving rational equations Solving rational equations, be sure to check your solutions.

1.

2.

3.

No solution

4.

5.

6.

is extraneous is extraneous

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Set Topic: Key features of rational functions For each of the given functions list the key features of the function including the domain, the intercepts, end behavior, and asymptotes.

7. ( )

8.

Domain: Domain:

Intercepts:

Intercepts:

End Behavior: End Behavior: Vertical Asymptote(s): Vertical Asymptote(s): Horizontal Asymptote: Horizontal Asymptote:

9.

10.

Domain: Domain: (

) (

)

Intercepts: None Intercepts: End Behavior: End Behavior:

Vertical Asymptote(s): Vertical Asymptote(s):

Horizontal Asymptote: Horizontal Asymptote:

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Sketch a graph and compare the given functions. Explain what you think accounts for the similarities and differences between the two functions.

11. a.

b.

| |

Similarities: Answers may vary…both graphs lie completely above the x-axis and have the same vertical asymptote. Same end behavior

Differences:

Answers may vary…Graph b has a faster rate of growth near the horizontal asymptote than graph a.

Why?

Answers may vary.

12. a.

b.

Similarities: Answers may vary…same end behavior, same horizontal & vertical asymptotes.

Differences:

Answers may vary. The middle sections of the graphs are different. Why?

Answers may vary.

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Topic: Identifying equations of slant asymptotes Find the equation of the slant asymptote for each rational function below.

13.

14.

15.

16.

Go Topic: Attributes of rational functions 17. How do you know if a function is even, odd, or neither?

Even functions: and contain y-axis symmetry Odd functions: and contains 180 symmetry about the origin.

18. How do you determine the end behavior of a rational function?

Compare the degrees of the numerator and denominator to determine the horizontal asymptotes. 19. Is the end behavior for a rational function always the same?

No 20. What is the difference between a proper and an improper rational function?

Proper: degree in the numerator is less than the degree in the denominator Improper: degree in the numerator is greater than or equal to the degree in the denominator

21. What attributes do all proper rational functions have in common?

Horizontal asymptote is at .

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Topic: Solving rational inequalities Solve each rational inequality. Write your answers in interval notation.

22.

23.

[

24. ( )

25.

] ] (

+

Documents

Name: Rational Functions 2.1H Readymsshultis.weebly.com/uploads/1/0/9/3/10930910/fall...1. 2. Yes Yes ( ) 3. 4. No Yes ( ) Set Topic: Simplifying rational expressions & operations