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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?
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?
How much would each person need to contribute if 45 people donate?
$75.00
What is the graph of the function xy + 25 = 0?
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?
What does the graph look like for the function y=8/x+3-2?
What is the equation of the function y=3/x translated 4 units to the right and 5 units down?
What is the equation of the function y=3/x translated 4 units to the right and 5 units down?
What are the points of discontinuity for the rational function y=(x-5)/(x^2-7x-8)?
What are the points of discontinuity for the rational function y=(x-5)/(x^2-7x-8)?
Describe the vertical asymptote(s) and hole(s) for the graph of y=(x+2)/(x^2+8x+15).
Describe the vertical asymptote(s) and hole(s) for the graph of y=(x+2)/(x^2+8x+15).
Find the horizontal asymptote of the graph of y=(-3x^5+5x+4)/(6x^5+2x+5).
Find the horizontal asymptote of the graph of y=(-3x^5+5x+4)/(6x^5+2x+5).
What is the graph of the rational function y=(x^2-8x+15)/(x^2-4)?
What is the graph of the rational function y=(x^2-8x+15)/(x^2-4)?
Simplify the rational expression t^2+3t-28/(t^2-16). State any restrictions on the variable.
Simplify the rational expression t^2+3t-28/(t^2-16). State any restrictions on the variable.
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).
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).
Which is equivalent to 3x/(x+5) - (x-5)/x?
Which is equivalent to 3x/(x+5) - (x-5)/x?
What is the difference in simplest form for (n^2+10n+21)/(n^2+3n-28) - (3n)/(n-4)?
What is the difference in simplest form for (n^2+10n+21)/(n^2+3n-28) - (3n)/(n-4)?
Simplify the sum (d^2-9n+18)/(d^2+2d-15) + (d^2-2d-8)/(d^2+5d+6).
Simplify the sum (d^2-9n+18)/(d^2+2d-15) + (d^2-2d-8)/(d^2+5d+6).
Simplify the complex fraction (n-7)/(n^2+10n+24)/(n-1)/(n+6).
Simplify the complex fraction (n-7)/(n^2+10n+24)/(n-1)/(n+6).
Solve the equation a/(a^2-64) + 3/(a-8) = 2/(a+8).
Solve the equation a/(a^2-64) + 3/(a-8) = 2/(a+8).
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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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