MATH Basic Algebra II  - Unit 24: Rational Expressions and Graphs

For responses that require non-keyboard symbols, the answers should be expressed in written form.  See examples below. 

For the following problems, “y varies directly as x”.  Write the appropriate direct variation equation and then solve for the missing variable.

Inverse, Joint, and Combined Variation

For the following problems, “y varies directly as x”.  For the given values of “x” and “y”, find the constant of variation, and then write the appropriate direct variation equation.  Use that equation to determine to solve for the missing value.

Hint: Use y = kx

1) Given:  y = 36 when x = 8.  If y varies directly as x, find y when x = 12.

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2)

Given:  x = 11 when y = 99.  If y varies directly as x, find x when y = 144

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3)

The softball team at your high school is selling pizza to raise money for new uniforms.  The amount of money raised by sales varies directly as the number of pizzas sold.  When 13 pizzas are sold, $94.25 is raised.  How many pizzas must be sold to raise $377? 

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For the following problems, “y varies inversely as x”.  For the given values of “x” and “y”, find the constant of variation, and then write the appropriate inverse-variation equation.  Use that equation to determine “y” for the second given x-value. 

4)

Given:  y = 12 when x = 7, find:  “y” when x = 5. 

5)

Given:  y = 0.4 when x = 2, find “y” when x = 0.1. 

6)

Given:  y = 12 when x = 0.4, find “y” when x = 0.5. 

7)

When two people are balanced on a seesaw, their distance from the center varies inversely as their weight.  If a 125 pound person sits 1.6 meters from the center of the seesaw, how far from the center should a 100 pound person sit?

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For the following problems, “y varies jointly as x and z”.  Write the appropriate joint-variation equation, and then find “y” for the given value of “x” and “z”. 

Hint:  Use  y = kxz

8)

Given:  y = 16 when x = 4 and z = 0.5, find “y” when x = 2 and z = 0.25. 

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9)

Given:  y = 120 when x = 2.5 and z = 2, find “y” when x = 3 and z = 2. 

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10)

Given:  y = 12 when x = 4 and z = 5, find “y” when x = 6 and z = 3. 

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11)

The volume (V) of a pyramid, varies jointly as its height (h) and the area of its Base (B).  A pyramid with a height measuring 12 inches and a base with area of 13 square inches has a volume of 52.   Find the volume of a pyramid having a  height of 7 inches and a base of area 36 square inches. 

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For the following problems, “z varies jointly as x and y and inversely as w”.  Write the appropriate combined-variation equation, and then find “z” for the given values of “x”, “y”, and “w”. 

12)

Given:  z = 320 when x = 4, y = 10, and w = 2.5, find “z” when x = 5, y = 6, and w = 8. 

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13)

Given:  z = 3.2 when x = 0.2, y = 8, and w = 4, find “z” when x = 3, y = 6, and w = 16. 

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Rational Functions and Their Graphs
For the following problems, determine whether the function is a rational function.  If so, find the domain.  If the function is not rational, explain why it is not. 

Hint: To find the domain:

1.)  Factor the denominator, if possible.
2.)  Use the zero product property to find values of x that = 0.
3.)  The values that you find in step 2 will be the values of x that are to be excluded from the solution.

14)

Rational function?  If so, find the domain;  if not, state why not. 

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15)

Rational function?  If so, find the domain;  if not, state why not. 

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16)

Rational function?  If so, find the domain;  if not, state why not. 

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For the following problems, identify all vertical and horizontal asymptotes of the graph of the rational function. 

17)

State the vertical and the horizontal asymptotes. 

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18)

State the vertical and the horizontal asymptotes. 

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19)

State the vertical and the horizontal asymptotes. 

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For the following problems,
            (a)  find the domain of the rational function,

            (b)  identify all asymptotes and

            (c)  identify any holes in the graph. 

20)

State the (a)  domain, (b)  vertical and horizontal asymptotes, and (c)  the holes in the graph, if any. 

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21)

State the (a)  domain, (b)  vertical and horizontal asymptotes, and (c)  the holes in the graph, if any. 

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Review

22)

Solve the absolute value inequality for x. 

Hint: There are two answers.

Click here to review the unit content explanation for Inequalities and Absolute Value Equations.

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23)

Rewrite the given expression using positive exponents. 

Click here to review the unit content explanation for Operations with Numbers and Exponents.

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24)

Factor the expression. 

Hint: Factor out the GCF

Click here to review the unit content explanation for Quadratic Functions.

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25)

Factor the expression. 

Click here to review the unit content explanation for Quadratic Functions.

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26)

Solve for x by factoring. 

Hint: Solve for zero first.

Click here to review the unit content explanation for Quadratic Functions.

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27)

Solve for x by factoring. 

Hint: Solve for zero first.

Click here to review the unit content explanation for Quadratic Functions.

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28)

Identify and state the vertex and axis of symmetry for the given quadratic function. 

Click here to review the unit content explanation for Solving Quadratic Equations.

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29)

Identify and state the vertex and axis of symmetry for the given quadratic function. 

Click here to review the unit content explanation for Solving Quadratic Equations.

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30)

State the degree of the polynomial. 

Click here to review the unit content explanation for Polynomial Functions.

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31) Extended Learning

Watch the following video, then write a five-sentence paragraph summarizing the video.

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