Algebra 2B Rational Functions Practice

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Questions and Answers

What is the equation that models the relationship if y varies jointly with w and x, inversely with z?

y=9wx/z

How much would each person need to contribute if 45 people donate?

$75.00

What is the graph of the function xy + 25 = 0?

Graph B (has rounded lines in the top left and bottom right corners)

What does the graph look like for the function y=8/x+3-2?

Graph D (curved lines with one towards the right going up and right, and one towards the left going left and down, with a center around -3, -2) Signup and view all the answers

What is the equation of the function y=3/x translated 4 units to the right and 5 units down?

y=3/x-4 -5 Signup and view all the answers

What are the points of discontinuity for the rational function y=(x-5)/(x^2-7x-8)?

x=-1, x=8 Signup and view all the answers

Describe the vertical asymptote(s) and hole(s) for the graph of y=(x+2)/(x^2+8x+15).

Asymptotes: x=-5, -3 and no holes Signup and view all the answers

Find the horizontal asymptote of the graph of y=(-3x^5+5x+4)/(6x^5+2x+5).

y=-1/2 Signup and view all the answers

What is the graph of the rational function y=(x^2-8x+15)/(x^2-4)?

Graph C (curved line in top left, capital L shaped line in top right, tall curved line in the bottom center) Signup and view all the answers

Simplify the rational expression t^2+3t-28/(t^2-16). State any restrictions on the variable.

t+7/(t+4), t≠4, t≠-4 Signup and view all the answers

What is the product in simplest form? State any restrictions on the variable for (x^2+9x+18)/(x+2)*(x^2-3x-10)/(2x+2-24).

(x+3)(x-5)/(x-4), x≠-6, x≠-2, x≠4 Signup and view all the answers

Which is equivalent to 3x/(x+5) - (x-5)/x?

2x^2+25/(x(x+5)) Signup and view all the answers

What is the difference in simplest form for (n^2+10n+21)/(n^2+3n-28) - (3n)/(n-4)?

-2n+3/(n-4) Signup and view all the answers

Simplify the sum (d^2-9n+18)/(d^2+2d-15) + (d^2-2d-8)/(d^2+5d+6).

2d^2-2d-38/((d+5)(d+3)) Signup and view all the answers

Simplify the complex fraction (n-7)/(n^2+10n+24)/(n-1)/(n+6).

n-7/((n+4)(n-1)) Signup and view all the answers

Solve the equation a/(a^2-64) + 3/(a-8) = 2/(a+8).

-20 Signup and view all the answers

Study Notes

Rational Functions Basics

Joint variation: ( y ) varies directly with ( w ) and ( x ), and inversely with ( z ).
Derived equation: ( y = \frac{9wx}{z} ) when values are substituted into the equation.

Inverse Variation in Donations

Inverse relationship: As the number of donors increases, the contribution per person decreases.
Calculation for 25 donors: Each contributes $135.
Contribution for 45 donors: $75.00 per person.

Graphing Functions

Equation ( xy + 25 = 0 ) results in a specific graph with defined features.
Asymptotes and graph behavior of ( y = \frac{8}{x} + 3 - 2 ): the graph shows a distinct curve with right and left movements.

Function Translations

Translated function from ( y = \frac{3}{x} ): a shift leads to new expression ( y = \frac{3}{x-4} - 5 ).

Points of Discontinuity

Rational function ( y = \frac{x-5}{x^2-7x-8} ) has discontinuities at ( x = -1 ) and ( x = 8 ).

Asymptotic Behavior

Vertical asymptotes identified for ( y = \frac{x+2}{x^2+8x+15} ): occur at ( x = -5 ) and ( x = -3 ) with no holes in the graph.
Horizontal asymptote for ( y = \frac{-3x^5 + 5x + 4}{6x^5 + 2x + 5} ) is ( y = -\frac{1}{2} ).

Analyzing Rational Functions Graphs

Rational function ( y = \frac{x^2 - 8x + 15}{x^2 - 4} ) produces a graph characterized by distinct shape patterns (curved lines and capital L shape).

Simplifying Rational Expressions

Simplified form of ( \frac{t^2 + 3t - 28}{t^2 - 16} ) yields ( \frac{t + 7}{t + 4} ) with restrictions ( t \neq 4, t \neq -4 ).
Product simplification for ( \frac{x^2 + 9x + 18}{x + 2} \cdot \frac{x^2 - 3x - 10}{2x + 2} - 24 ) results in ( \frac{(x + 3)(x - 5)}{x - 4} ) with restrictions ( x \neq -6, x \neq -2, x \neq 4 ).

Equivalence and Differences

Simplified expression of ( \frac{3x}{x + 5} - \frac{x - 5}{x} ) leads to ( \frac{2x^2 + 25}{x(x+5)} ).
Difference of two rational expressions ( \frac{n^2 + 10n + 21}{n^2 + 3n - 28} - \frac{3n}{n - 4} ) results in ( \frac{-2n + 3}{n - 4} ).

Complex Fractions

Simplified complex fraction ( \frac{n - 7}{\frac{n^2 + 10n + 24}{n - 1}} \div (n + 6) ) results in ( \frac{n - 7}{(n + 4)(n - 1)} ).

Solving Rational Equations

Solution to the equation ( \frac{a}{a^2 - 64} + \frac{3}{a - 8} = \frac{2}{a + 8} ) is ( a = -20 ).

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Description

This quiz focuses on rational functions, specifically how to model relationships involving joint and inverse variation. Test your understanding by solving problems related to real-life situations, such as fundraising and playground construction. Master the concepts to excel in Algebra 2B.