Algebra IIA
Unit V: Rational and Radical Functions
• Foundational Material
o Solve problems with linear functions
o Graph functions with asymptotes
o Simplify polynomial expressions
o Solve quadratic equations and inequalities
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Goal
Solve problems with variable functions
 Solve rational and radical equations and
inequalities
Simplify rational and radical expressions
Graph radical and rational functions
Why?
To further build a foundation for higher level mathematics such as pre-calculus, statistics, and business calculus
To solve problems in other classes such as chemistry, physics, and biology
Make predictions involving time, money and speed
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Key Vocabulary
Complex Fraction
Constant of Variation
Continuous Function
Direct Variation
Discontinuous Function
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Extraneous Solutions
Hole in a Graph
Inverse Variation
Inverse Function
Radical Equation
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Radical Function
Rational Equation
Rational Exponent
Rational Function
Lesson I: Variation Functions and Multiplying and Dividing Rational Expressions
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Solve problems involving direct, inverse, joint, and combined variation.
Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales. (CC.9-12.A.CED.2)
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities,
and interpret solutions as viable or nonviable options in a modeling context.
Simplify rational expressions.
Multiply and divide rational expressions.
Understand that rational expressions form a system analogous to the rational numbers, closed under
addition, subtraction, multiplication, and division by nonzero rational expression; add, subtract, multiply,
and divide rational expressions. (CC.9-12.A.APR.7)
Direct Variation: ________________ (y varies directly as x, k is the constant of variation and )
Joint Variation: __________________ (y varies jointly as x and z, k is the constant of variation and )
Inverse Variation: ________________ (y varies inversely as x, k is the constant of variation and )
Examples:
1. Given that y varies directly as x, and y = 6.5
when x = 13 write and graph the direct
variation function.
2.
Given that y varies directly as x, and
y = 14 when x = 3.5. Write and graph
the direct variation function.
3. The circumference of a circle C varies directly
as the radius r, and C = 𝜋 ft when r = 3.5 ft.
Find r when C = 2𝜋ft.
6. The lateral surface area L of a cone varies
jointly as the base radius r and the slant height s
and L = 63 m2 when r = 3.5 m and s = 18 m. Find
r to the nearest tenth when L = 8 m2 and s = 5
m.
7. Given that y varies inversely as x and y = 2
when x = 3. Write and graph the inverse
variation function.
4. The perimeter P of a regular dodecagon varies
directly as the side length s and P =18 in. when
s = 1.5 in. Find s when P = 75 in.
8. Given that y varies inversely as x and y = 1/5
when x = 10 write and graph the inverse
variation function.
5. The area A of a triangle varies jointly as the
base b and the height h, and A = 12 m2 when b
= 6 m and h = 4 m. Find b when A = 36 m2
and h = 8 m.
9. Given that y varies inversely as x and y = 12
when x = 1/4, write and graph the inverse
variation function.
10. The time it takes to construct a house varies
inversely as the number of volunteers v. If 20
volunteers can build a house in 62.5 working
hours, how many volunteers would be needed
to build a house in 50 working hours?
11. The time t needed to complete a certain race
varies inversely as the runner's average speed s.
If a runner with an average speed of 8.82 mph
completes the race in 2.97 hours, what is the
average speed of a runner who completes the
race in 3.5 hours?
Determining Type of Variation from a Table
X
6
12
18
24
30
36
42
y
120
60
40
30
24
20
120/7
What variation model does the table represent (direct, If the table represents direct or inverse variation, what
inverse or neither)?
is the equation that models the table?
X
6.5
13
104
X
1
40
26
y
8
4
0.5
y
0.2
8
5.2
What variation model does the table represent?
What is the equation that models the table?
X
5
8
12
y
30
48
72
What variation model does the table?
What is the equation that models the table?
What variation model does the table represent?
What is the equation that models the table?
X
3
6
8
y
5
14
21
What variation model does the table represent?
What is the equation that models the table?
Simplifying, Multiplying, and Dividing Rational Expressions
A rational expression is a ______________________________________________________________.
Rational expressions can be simplified by _________________________________________________.
Simplify.
3x 7
2x 4
x 2 - 2x - 3
x 2 + 5x + 4
16x11
8x 2
3x + 4
3x 2 + x - 4
6x 2 + 7x + 2
6x 2 - 5x - 6
2x - x 2
x2 - x - 2
To multiply rational expressions write the problem as a single fraction then factor and reduce.
x2 x4
 2
3x  12 x  4
2 x 4 y 5 15x 2
 3 2
2
3x
8x y
10 x  40
x 3
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x 2  6 x  8 5 x  15
To divide rational expressions FIRST rewrite the expression as a multiplication problem, then write the problem
as a single fraction and then factor and reduce.
4 x3
16
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9 x2 y 9 y5
x5  4 x3 x5  x 4  2 x3
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x2  x  2
x2 1
2x2  7 x  4
4x2 1
 2
x2  9
8x  28x  12
You try:
4 x6
2x  6
6 x 2  13x  5
6 x 2  23x  7
x 2  16
x2
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x2  4 x  4 x2  6 x  8
x2  2 x  1
x2 1
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x 2  3x  18 x 2  7 x  6
Assignment: Page 318, 17-35 odd, 46 and Page 324, 19-31 odd