TE-16
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
polynomial.
SDUHSD Math 3 Honors
is not a polynomial. √
isn’t rational because it isn’t a
TE-17
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
should be:
terms in the 2nd line when she needed a common denominator. Answer
.
Simplify each expression. Leave answers in factored form.
8.
9.
10.
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.
11.
TE-18
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
b. The inequality
inequality
c.
d.
is
can be solved by multiplying both sides by
.
and
has no real solution.
SDUHSD Math 3 Honors
.
have the same solution set.
, resulting in the equivalent
TE-19
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.
SDUHSD Math 3 Honors
TE-20
Topic: Trigonometric ratios in a right triangle
Complete the given trigonometric ratios for
28.
SDUHSD Math 3 Honors
.
29.
TE-30
Name:
Rational Functions
2.2H
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.
9.
8.
proper
improper,
11.
10.
improper,
SDUHSD Math 3 Honors
improper,
12.
improper,
proper
TE-31
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:
Domain:
Asymptotes: None
15.
Zeros:
Domain:
Asymptotes: None
16.
Zeros:
Domain:
Asymptotes:
17.
√
Zeros:
Domain:
Asymptotes: None
SDUHSD Math 3 Honors
Zeros:
Domain:
Asymptotes:
18.
Zeros:
Domain:
Asymptotes: None
TE-32
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.).
List of Features:
Domain:
19.
Range:
Continuous: not continuous
Intercepts: none
Asymptotes:
0
Und
2
1
End behavior:
As
As
1
2
List of Features:
Domain:
20.
Range:
Continuous: not continuous
Intercepts: no x-intercepts
y-intercept: ( )
Asymptotes:
Und
2
1
0
SDUHSD Math 3 Honors
End behavior:
As
As
TE-33
List of Features:
Domain:
21.
Range:
Continuous: not continuous
Intercepts: none
Asymptotes:
0
Und
End behavior:
As
As
6
1
3
2
List of Features:
Domain:
22.
Range:
Continuous: not continuous
1
Intercepts: none
2
0
Asymptotes:
Und
End behavior:
As
As
1
23. What happens to
The values of
as the x-values get closer to zero in the graph in question 22?
approach
SDUHSD Math 3 Honors
TE-34
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.
(
)
SDUHSD Math 3 Honors
+
TE-43
Name:
Rational Functions
2.3H
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.
Improper
4.
3.
Improper
Proper
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
8.
9.
and
.
10.
as
as
as
as
as
as
SDUHSD Math 3 Honors
TE-44
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
15.
Proper
16.
Improper
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Improper
TE-45
Go
Topic: Simplifying rational expressions
Simplify each of the rational expressions by canceling common factors. Leave answers in factored
form, where possible.
17.
18.
20.
19.
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.
[
SDUHSD Math 3 Honors
[
]
TE-57
Name:
Rational Functions
2.4H
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.
2.
Domain:
Domain: [
Range:
]
Range:
]
3.
4.
Domain: [
Range: [
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
5.
6.
As
As
As
As
SDUHSD Math 3 Honors
and also
.
TE-58
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
SDUHSD Math 3 Honors
TE-59
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.
even
e.
f.
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
SDUHSD Math 3 Honors
18. Odd Function
TE-60
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.
(
)
SDUHSD Math 3 Honors
)
TE-70
Name:
Rational Functions
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
SDUHSD Math 3 Honors
4.
Perimeter:
Area:
2.5H
TE-71
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
as
End Behavior: as
as
Vertical Asymptote(s):
Vertical Asymptote(s):
Horizontal Asymptote:
Horizontal Asymptote:
Sign Line:
Sign Line:
Graph:
Graph:
SDUHSD Math 3 Honors
TE-72
7.
8.
Domain:
Domain:
Intercepts: x-int: 2, y-int: 2
Intercepts: None
End Behavior: as
as
End Behavior: as
as
Vertical Asymptote(s):
Vertical Asymptote(s):
Horizontal Asymptote:
Horizontal Asymptote:
Sign Line:
Sign Line:
Graph:
Graph:
SDUHSD Math 3 Honors
TE-73
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.
13.
14.
)
12.
Topic: Solving rational equations
Solve each equation.
15.
√
17.
SDUHSD Math 3 Honors
16.
is extraneous
TE-85
Name:
Rational Functions
2.6H
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
SDUHSD Math 3 Honors
is extraneous
TE-86
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:
SDUHSD Math 3 Honors
)
(
)
TE-87
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.
SDUHSD Math 3 Honors
TE-88
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
.
SDUHSD Math 3 Honors
TE-89
Topic: Solving rational inequalities
Solve each rational inequality. Write your answers in interval notation.
22.
23.
[
24.
(
)
]
25.
]
SDUHSD Math 3 Honors
(
+