Chapter 8: Q. 11 (page 556)

Analyze the graph of the rational function
$R (x) = \frac{2 x^{2} - 7 x - 4}{x^{2} + 2 x - 15}$

Short Answer

Expert verified

The domain of the function is $\{x | x \neq - 5, x \neq 3\}$ .

Its intercepts are $(\frac{- 1}{2}, 0), (4, 0), (0, \frac{4}{15})$ .

The asymptotes of the function are at $y = - 5, y = 3, x = 2$

Step by step solution

Step 1. Given Information

The given expression is $R (x) = \frac{2 x^{2} - 7 x - 4}{x^{2} + 2 x - 15}$ .

Step 2. Find the Domain

Find the factors of the expression in the numerator and the denominator.

$R (x) = \frac{(2 x + 1) (x - 4)}{(x + 5) (x - 3)}$

The denominator will become 0 if the value of xis $- 5$ or 3.
So, the domain of the function is $\{x | x \neq - 5, x \neq 3\}$ .

Step 3. Find The Intercepts

The numerator of the function becomes 0 when the value of xis $\frac{- 1}{2}, 4$ .
So, the x-intercepts are $(- \frac{1}{2}, 0), (4, 0)$ .
The value of $R (0) = \frac{4}{15}$ /
So, the y-intercept is $(0, \frac{4}{15})$ .

Step 4. Find the Asymptotes

The denominator becomes 0 at $- 5, 3$ .
So, the vertical asymptotes are $x = - 5, x = 3$ .
The ratio of leading coefficients of the numerator and denominator is 2.
So, the vertical asymptote is $y = 2$ .

Step 5. Plot the Graph

Use the information above to plot the following graph.

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Analyze the graph of the rational function
$R (x) = \frac{2 x^{2} - 7 x - 4}{x^{2} + 2 x - 15}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Domain

Step 3. Find The Intercepts

Step 4. Find the Asymptotes

Step 5. Plot the Graph

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Analyze the graph of the rational functionRx=2x2-7x-4x2+2x-15

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the Domain

Step 3. Find The Intercepts

Step 4. Find the Asymptotes

Step 5. Plot the Graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

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Pure Maths

Study anywhere. Anytime. Across all devices.

Analyze the graph of the rational function
$R (x) = \frac{2 x^{2} - 7 x - 4}{x^{2} + 2 x - 15}$