 ## MATH1111 Quizzes

Rational Functions Quiz
Web resources available Questions

This quiz tests the work covered in Lecture 6 and corresponds to the second half of Section 1.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are further web quizzes at Wiley. Choose section 5 from this page.

Be aware that it doesn’t seem to accept the written answers so you will have to check whether your answers are correct when they print the correct answer. Question 15 didn’t make sense on 7/11/05.

There is a written explanation at http://id.mind.net/ zona/mmts/functionInstitute/rationalFunctions/rationalFunctions.html which covers the material in the text but it also has links to a variety of applets which are all very useful.

If you are still not sure how to draw these graphs yourself you may want to look at http://www.math.csusb.edu/math110/src/rationals/RfIntro.html and after the demo there is a step by step guide to graphing rational functions. It also has an exercise where you get to work out where the appropriate features of the graph lie.

Consider $f\left(x\right)=\frac{5{x}^{3}-{x}^{2}-4}{2{x}^{3}+{x}^{2}-1}\phantom{\rule{0.3em}{0ex}}.$
Which of the following set of statements is correct? Exactly one option must be correct)
a)
 $f\left(x\right)\to \frac{5}{2}$ as $x\to -\infty$ $f\left(x\right)\to \frac{5}{2}$ as $x\to +\infty$
b)
 $f\left(x\right)\to -\frac{5}{2}$ as $x\to -\infty$ $f\left(x\right)\to \frac{5}{2}$ as $x\to +\infty$
c)
 $f\left(x\right)\to -\infty$ as $x\to -\infty$ $f\left(x\right)\to \infty$ as $x\to +\infty$
d)
 $f\left(x\right)\to \infty$ as $x\to -\infty$ $f\left(x\right)\to \infty$ as $x\to +\infty$

Choice (a) is correct!
$f\left(x\right)\approx \frac{5{x}^{3}}{2{x}^{3}}=\frac{5}{2}$ for large positive $x$ and large negative $x\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Try again, $f\left(x\right)\approx \frac{5{x}^{3}}{2{x}^{3}}=\frac{5}{2}$ for large positive $x$ and large negative $x\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is incorrect
Try putting some large positive numbers and large negative numbers in to the function and see what happens.
Choice (d) is incorrect
Try putting some large positive numbers and large negative numbers in to the function and see what happens.
Which of the following are the $x$-intercepts for the function below?
$y=\frac{3{x}^{2}-12x+9}{{x}^{2}+5x+6}\phantom{\rule{0.3em}{0ex}}.$ Exactly one option must be correct)
 a) $x=-3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=-2\phantom{\rule{0.3em}{0ex}}.$ b) $x=-3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=-1\phantom{\rule{0.3em}{0ex}}.$ c) $x=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=9\phantom{\rule{0.3em}{0ex}}.$ d) $x=3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=1$

Choice (a) is incorrect
Try again, you have factorized the denominator, not the numerator.
Choice (b) is incorrect
Try again, you may not have factorized the numerator correctly.
Choice (c) is incorrect
Try again, you have not factorized the numerator correctly.
Choice (d) is correct!
$y=\frac{3\left({x}^{2}-4x+3\right)}{{x}^{2}+5x+6}=\frac{3\left(x-3\right)\left(x-1\right)}{{x}^{2}+5x+6}\phantom{\rule{0.3em}{0ex}}.$
Hence $y=0$ when $x=3$ or $x=1$ and these are the $x$-intercepts.
Which of the following are the vertical asymptotes for the function below?
$y=\frac{3{x}^{2}-12x+9}{{x}^{2}+5x+6}\phantom{\rule{0.3em}{0ex}}.$ Exactly one option must be correct)
 a) $x=-3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=-2\phantom{\rule{0.3em}{0ex}}.$ b) $x=3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=2\phantom{\rule{0.3em}{0ex}}.$ c) $x=3\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=1\phantom{\rule{0.3em}{0ex}}.$ d) $x=-6\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}x=1\phantom{\rule{0.3em}{0ex}}.$

Choice (a) is correct!
$y=\frac{3\left({x}^{2}-4x+3\right)}{{x}^{2}+5x+6}=\frac{3\left(x-3\right)\left(x-1\right)}{\left(x+3\right)\left(x+2\right)}\phantom{\rule{0.3em}{0ex}}.$
Hence $y$ is undefined when $x=-3$ or $x=-2$ and these are the vertical asymptotes.
Choice (b) is incorrect
Try again, you may not have factorized the denominator correctly.
Choice (c) is incorrect
Try again, you may have factorized the numerator, not the denominator.
Choice (d) is incorrect
Try again, you may not have factorized the denominator correctly.
Note that ${x}^{2}+5x+6=\left(x+3\right)\left(x+2\right)\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the horizontal asymptote for   $y=\frac{5{x}^{2}-8x+3}{3{x}^{2}+13x+4}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) $x=-1$ b) $y=5$ c) $y=\frac{5}{3}$ d) Since the numerator and the denominator cannot be factorized we cannot determine the horizontal asymptote.

Choice (a) is incorrect
Try again, this is one of the vertical asymptotes.
Remember horizontal asymptotes are of the form $y=b\phantom{\rule{0.3em}{0ex}}.$
Choice (b) is incorrect
Try again, look at what happens for large values of $x\phantom{\rule{0.3em}{0ex}}.$
Choice (c) is correct!
$f\left(x\right)\approx \frac{5{x}^{2}}{3{x}^{2}}=\frac{5}{3}$ for large positive $x$ and large negative $x\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Try again, the numerator and the denominator can be factorized,
but even if they couldn’t be factorized it would make no difference,
we could still find the horizontal asymptote.