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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
A

Recall / Warm-Up

1.

Which of the following shows the correct simplification of x2−9x−3\frac{x^2 - 9}{x - 3}?

2.

For the polynomial p(x)=x2−5x+6p(x) = x^2 - 5x + 6, which values of xx are zeros?

3.

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

B

Fluency Practice

1.

Find all zeros of f(x)=x2−4x2+1f(x) = \frac{x^2 - 4}{x^2 + 1}.

Enter both zeros separated by a comma (e.g., -3, 3).

2.

Let g(x)=x2−x−6x2−4g(x) = \frac{x^2 - x - 6}{x^2 - 4}. After factoring and simplifying, which describes the key features?

3.

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

4.

What is the horizontal asymptote of r(x)=5x2−3x2x2+7r(x) = \frac{5x^2 - 3x}{2x^2 + 7}?

5.

Which rational function has NO horizontal asymptote?

C

Varied Practice

1.

For f(x)=(x−1)(x+4)(x−1)(x+2)f(x) = \frac{(x-1)(x+4)}{(x-1)(x+2)}, the zero(s) of ff are:

2.

Let f(x)=x2−4(x−2)(x+3)f(x) = \frac{x^2 - 4}{(x-2)(x+3)}. Which correctly identifies all vertical asymptotes and holes?

3.

As x→∞x \to \infty, what does f(x)=3x+2x2−1f(x) = \frac{3x + 2}{x^2 - 1} approach?

4.

Consider f(x)=x−1x+1f(x) = \frac{x - 1}{x + 1} with horizontal asymptote y=1y = 1. Which statement is true?

5.

For f(x)=x2−9(x−3)(x+5)f(x) = \frac{x^2 - 9}{(x-3)(x+5)}, factor the numerator to get (x−___)(x+___)(x - \_\_\_)(x + \_\_\_). The shared factor cancels, creating a   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   at x=3x = 3. The remaining denominator factor creates a vertical   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   at x=−5x = -5.

first factor zero:
second factor zero:
feature type at x=3:
feature type at x=-5:
D

Word Problems

1.

A company's average cost per unit (in dollars) when producing xx units is C(x)=5000+4xxC(x) = \frac{5000 + 4x}{x}, where x>0x > 0.

1.

What is the horizontal asymptote of C(x)C(x), and what does it represent in context?

2.

What is the average cost per unit when the company produces 500 units?

2.

A rational function is given in factored form: f(x)=(x+2)(x−5)(x+2)(x−1)(x+4)f(x) = \frac{(x+2)(x-5)}{(x+2)(x-1)(x+4)}.

After simplifying, how many vertical asymptotes does ff have? Enter the count as a whole number.

E

Error Analysis

1.

A student analyzed f(x)=xx2−1f(x) = \frac{x}{x^2 - 1} and wrote:

"The denominator x2−1=(x−1)(x+1)x^2 - 1 = (x-1)(x+1) equals zero at x=1x = 1 and x=−1x = -1. These are vertical asymptotes. Since the graph has vertical asymptotes, it can still pass through x=1x = 1 if the yy-value is very large."

What error did the student make?

2.

A student found the HA of r(x)=3x2−2x+14x2+x−5r(x) = \frac{3x^2 - 2x + 1}{4x^2 + x - 5} and wrote:

"Both degrees are 2. But I also averaged in the −2x-2x and +x+x terms. After accounting for all terms, I get the HA is around y=25y = \frac{2}{5}."

What is the correct horizontal asymptote, and what did the student do wrong?

F

Challenge / Extension

1.

Consider f(x)=(x−1)(x+3)(x+3)(x−4)f(x) = \frac{(x-1)(x+3)}{(x+3)(x-4)}.

(a) Without graphing technology, identify all zeros, vertical asymptotes, holes, and the horizontal asymptote.
(b) Determine the sign of f(x)f(x) in each region between the key features and describe the overall graph shape.

2.

Design a rational function f(x)f(x) with all of these features: a zero at x=2x = 2, a vertical asymptote at x=−3x = -3, a hole at x=1x = 1, and a horizontal asymptote at y=2y = 2. Explain how you built your function and verify each feature.

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