Notes
Working with rational expressions
- Simplify by factoring and canceling common factors. State the excluded values from the original denominator.
- Multiply across; divide by multiplying by the reciprocal.
- To add or subtract, rewrite with the LCD, combine numerators carefully, then simplify.
Graphs
- A vertical asymptote is at a zero of the denominator that does not cancel. A canceled factor leaves a hole.
- Horizontal asymptote: lower degree on top gives $y = 0$; equal degrees give the ratio of leading coefficients; higher degree on top gives none (a slant asymptote if only one higher).
Equations and models
- Solve rational equations by multiplying by the LCD, then check every answer. Reject any that make a denominator 0 (extraneous).
- Work problems: $\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$. Inverse variation: $xy = k$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \frac{(x - 4)(x + 4)}{(x - 4)(x + 3)} = \frac{x + 4}{x + 3}$.
$\displaystyle \frac{3(x + 1) + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}$.
Set $2x - 6 = 0$, so $x = 3$.
The numerator has lower degree than the denominator.
Multiply by $x - 4$: $x = 4 + 2(x - 4)$, so $x = 2x - 4$ and $x = 4$. But $x = 4$ makes a denominator 0, so it is extraneous.
$\displaystyle \frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ of the room per hour.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A club sells T-shirts. There is a $200 setup fee and each shirt costs $8 to print. The average cost per shirt for $n$ shirts is $A(n) = \frac{200 + 8n}{n}$.
(a) Find the average cost per shirt when 40 shirts are printed.
Show the answer and scoring guide
$\displaystyle A(40) = \frac{200 + 320}{40} = \frac{520}{40} = 13$. The average cost is $13 per shirt.
Check off the points you earned:
(b) Find the horizontal asymptote of $A(n)$ and explain what it means.
Show the answer and scoring guide
The degrees are equal, so $\displaystyle y = \frac{8}{1} = 8$. As more shirts are printed, the average cost per shirt gets closer to $8, the printing cost per shirt, because the setup fee is spread over more shirts.
Check off the points you earned:
(c) How many shirts must be printed for the average cost to be $10?
Show the answer and scoring guide
$\displaystyle \frac{200 + 8n}{n} = 10 \Rightarrow 200 + 8n = 10n \Rightarrow n = 100$.
Check off the points you earned: