Notes
Inverse variation
$\displaystyle y = \frac{k}{x}$ (or $xy = k$). Find $k$ from one known pair, then use it for the next.
Working with rational expressions
- Simplify: factor fully, then cancel common factors. Note excluded values.
- Multiply: factor, cancel, multiply. Divide: multiply by the reciprocal.
- Add/subtract: rewrite over the least common denominator, then combine numerators.
Graph features
| Feature | How to find it |
|---|---|
| Hole | factor cancels from top and bottom |
| Vertical asymptote | zero of the simplified denominator |
| Horizontal asymptote | degree top $<$ bottom: $y = 0$; equal: ratio of leading coefficients; top $>$ bottom: none |
Solving rational equations
Multiply by the LCD, solve, then throw out any answer that makes an original denominator 0. For work problems, add the rates: $\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$k = 36$, so $\displaystyle 9 = \frac{36}{x}$ and $x = 4$.
$\displaystyle \frac{(x - 3)(x + 2)}{(x - 3)(x - 1)}$; cancel $x - 3$.
The denominator $(x - 1)(x + 1)$ has two zeros, and the bottom has higher degree.
$\displaystyle \frac{2(x - 3) - (x + 3)}{(x + 3)(x - 3)} = \frac{x - 9}{x^2 - 9}$.
Cross multiply: $4(x - 1) = 2(x + 1) \Rightarrow 2x = 6$.
$\displaystyle \frac{1}{4} + \frac{1}{t} = \frac{1}{2.4} = \frac{5}{12}$, so $\displaystyle \frac{1}{t} = \frac{2}{12}$ and $t = 6$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A club prints T-shirts. Setup costs $500 and each shirt costs $4 to print. The average cost per shirt for $x$ shirts is $A(x) = \frac{500 + 4x}{x}$ dollars.
(a) Find the average cost per shirt for 100 shirts.
Show the answer and scoring guide
$\displaystyle A(100) = \frac{500 + 400}{100} = 9$, so $9 per shirt.
Check off the points you earned:
(b) How many shirts must be printed for the average cost to be $6 per shirt?
Show the answer and scoring guide
$\displaystyle \frac{500 + 4x}{x} = 6 \Rightarrow 500 + 4x = 6x \Rightarrow x = 250$ shirts.
Check off the points you earned:
(c) Find the horizontal asymptote of $A(x)$ and explain what it means.
Show the answer and scoring guide
The top and bottom both have degree 1, so the asymptote is $\displaystyle y = \frac{4}{1} = 4$. As more shirts are printed, the average cost gets closer to $4 per shirt but never reaches it, because the $500 setup is spread over more shirts.
Check off the points you earned: