Back to Exercise: Graph rational functions

Exercises: Graph Rational Functions

Work through each section in order. Show your work where indicated. Factor all rational expressions completely before identifying key features.

Grade 9·20 problems·~30 min·Common Core Math - HS Functions·standard·hsf-if-c-7d

Work through problems with immediate feedback

Recall / Warm-Up

Which of the following shows the correct simplification of $\frac{x^2 - 9}{x - 3}$ ?

$x + 3$ for all $x$

$x + 3$ for $x \neq 3$

$x - 3$ for $x \neq 3$

$\frac{1}{x - 3}$

For the polynomial $p(x) = x^2 - 5x + 6$ , which values of $x$ are zeros?

$x = 2$ and $x = 3$

$x = -2$ and $x = -3$

$x = 5$ and $x = 6$

$x = 1$ and $x = 6$

The expression $\frac{1}{x - 4}$ is undefined when $x$ equals which value?

$x = 0$

$x = 1$

$x = 4$

$x = -4$

Fluency Practice

Find all zeros of $f(x) = \frac{x^2 - 4}{x^2 + 1}$ .

The smaller zero is $x =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ and the larger zero is $x =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

smaller zero:

larger zero:

Let $g(x) = \frac{x^2 - x - 6}{x^2 - 4}$ . After factoring and simplifying, which describes the key features?

Zero at $x = 3$ ; vertical asymptote at $x = 2$ ; hole at $x = -2$

Zeros at $x = 3$ and $x = -2$ ; vertical asymptotes at $x = 2$ and $x = -2$

Zero at $x = 3$ ; vertical asymptotes at $x = 2$ and $x = -2$ ; no holes

Zero at $x = -2$ ; vertical asymptote at $x = 3$ ; hole at $x = 2$

For $h(x) = \frac{2x}{(x+1)(x-3)}$ , which statement about vertical asymptotes is correct?

There are no vertical asymptotes.

Vertical asymptote at $x = 0$ only.

Vertical asymptotes at $x = -1$ and $x = 3$ ; the graph approaches but never crosses these lines.

Vertical asymptotes at $x = -1$ and $x = 3$ ; the graph may cross these lines.

What is the horizontal asymptote of $r(x) = \frac{5x^2 - 3x}{2x^2 + 7}$ ?

$y = 0$

$y = \frac{5}{2}$

$y = -\frac{3}{7}$

No horizontal asymptote

Which rational function has NO horizontal asymptote?

$f(x) = \frac{3}{x^2 + 1}$

$g(x) = \frac{4x - 1}{2x + 3}$

$h(x) = \frac{x^2 + x}{x - 2}$

$k(x) = \frac{x + 7}{x^2 - 9}$

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