Precalculus Rational Functions and Asymptotes Test Solutions and Scoring (PDF)

Precalculus math test solutions: Rational Functions and Asymptotes – Maths PDF

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This rational functions test answers for Precalculus covers rational functions and asymptotes (Common Core A-APR.D.7, F-IF.C.7d, A-CED.A.1), with a free PDF to print.

Worked solutions and scoring notes for the 9-question rational functions test. Use the scoring line under each solution to award partial credit.

Precalculus · Common Core State Standards: A-APR.D.7, F-IF.C.7d, A-CED.A.1

Rational functions test answers: worked solutions

Question 1 – Domain and vertical asymptotes [2 pts]

$x^{2}-9=0$ at $x=\pm3$. Neither cancels with $x-4$.

Domain $x\neq\pm3$; asymptotes $x=3$ and $x=-3$

Scoring: 1 point for the domain, 1 point for the asymptotes.

Question 2 – Horizontal asymptotes [2 pts]

(a) Equal degrees: $\frac52$. (b) Lower numerator degree: 0.

(a) $y=\frac52$; (b) $y=0$

Scoring: 1 point each.

Question 3 – Hole and asymptote [2 pts]

$f(x)=\frac{(x-3)(x+2)}{(x-3)(x+1)}$. Hole at $x=3$, $y=\frac{5}{4}$. Asymptote $x=-1$.

Hole $\left(3,\frac54\right)$; asymptote $x=-1$

Scoring: 1 point for the hole (both coordinates), 1 point for the asymptote.

Question 4 – Slant asymptote [2 pts]

Synthetic division with $c=-1$: $2,3,-1\to 2,1,-2$. The quotient is $2x+1$ with remainder $-2$.

$y=2x+1$

Scoring: 1 point for dividing, 1 point for the line.

Question 5 – Read a rational graph [3 pts]

Asymptotes x = -1 and y = 2.

(a) $x=-1$ and $y=2$. (b) $(1,0)$ and $(0,-2)$. (c) The domain excludes $-1$; the range excludes $2$.

$x=-1,\ y=2$; $(1,0),(0,-2)$; domain $x\neq-1$, range $y\neq2$

Scoring: 1 point for each part.

Question 6 – Solve a rational equation [2 pts]

$4(x-2)=3(x+1)$, $4x-8=3x+3$, $x=11$. Check: $\frac{4}{12}=\frac13$ and $\frac{3}{9}=\frac13$.

$x=11$

Scoring: 1 point for clearing the fractions, 1 point for the check and answer.

Question 7 – Extraneous solution [2 pts]

$x=2+3(x-2)$, so $x=3x-4$ and $x=2$. But $x=2$ makes the denominators zero.

No solution

Scoring: 1 point for solving, 1 point for rejecting the extraneous value.

Question 8 – Average cost [3 pts]

At n = 40 the average is 35 dollars.

(a) $\frac{800+600}{40}=35$. (b) $y=15$: the average approaches the per-item cost. (c) $800+15n=20n$, $n=160$.

$\$35$; $y=15$; $160$ bracelets

Scoring: 1 point for each part.

Question 9 – Rational inequality [2 pts]

Critical values $-3$ and $1$. Test $x=0$: $\frac{3}{-1}< 0$. Test $x=-4$: $\frac{-1}{-5}>0$. Test $x=2$: $\frac{5}{1}>0$.

$(-3,1)$

Scoring: 1 point for the sign test, 1 point for the interval.

Total score

Total

20 points

Key facts behind this rational functions test answers

Key facts

Domain: all real numbers except zeros of the denominator.
Vertical asymptote: a denominator zero that does not cancel. Hole: a cancelled factor.
Degree of numerator < denominator: y = 0. Equal degrees: y = ratio of leading coefficients.
Degree of numerator = degree of denominator + 1: slant asymptote from division.
Always check rational equation answers for extraneous solutions.

How to use this rational functions test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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