Chapter 4: Q. 15 (page 234)

In Problems $7 - 44$ , follow Steps $1$ through $7$ on page $228$ to analyze the graph of each function.
$H (x) = \frac{x^{3} - 1}{x^{2} - 9}$

Short Answer

Expert verified

The graph of the function $H (x) = \frac{x^{3} - 1}{x^{2} - 9}$ is,

Step by step solution

Step 1 Given function is,

$H (x) = \frac{x^{3} - 1}{x^{2} - 9}$

which may also written as,

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

The domain of $H$ is $\{x | x \neq - 3, x \neq 3\}$

So, there will be domain in $0$ .

Substitute $x = 0$ in the given function.

$H (0) = \frac{0 - 1}{0 - 9} H (0) = \frac{1}{9}$

So, the obtained point is, $(0, \frac{1}{9})$ .

Step 2 The function H is in lower terms.

The real zeros of the denominator are the real solutions of the equation.

$(x + 3) (x - 3) = 0 x = 3, - 3$

The two lines $x = 3, x = - 3$ are two vertical asymptotes of the graph.

Graph these asymptotes using the dashed lines.

Step 3 In the given function degree of the numerator is greater than the degree of the denominator.

Use the method of long division to find the horizontal asymptote.

Therefore, $H (x) = x + \frac{9 x - 1}{(x + 3) (x - 3)}$ .

The quotient is $x$ . So, the line $y = x$ is an oblique asymptote of the graph. Indicate the line $y = x$ on the graph using a dashed line.

Step 4 Substitute Hx=x to determine whether the graph intersects at the asymptote y=x.

$x = \frac{x^{3} - 1}{x^{2} - 9} x (x^{2} - 9) = x^{3} - 1 x^{3} - 9 x = x^{3} - 1 - 9 x = - 1 x = \frac{1}{9}$

As $y = x$ , $y = \frac{1}{9}$ .

So, the point of intersection is, $(\frac{1}{9}, \frac{1}{9})$ .

Step 5 Now plot a graph by using the known values.

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In Problems $7 - 44$ , follow Steps $1$ through $7$ on page $228$ to analyze the graph of each function.
$H (x) = \frac{x^{3} - 1}{x^{2} - 9}$

Short Answer

Step by step solution

Step 1 Given function is,

Step 2 The function H is in lower terms.

Step 3 In the given function degree of the numerator is greater than the degree of the denominator.

Step 4 Substitute Hx=x to determine whether the graph intersects at the asymptote y=x.

Step 5 Now plot a graph by using the known values.

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In Problems 7-44, follow Steps 1through 7on page 228to analyze the graph of each function.Hx=x3-1x2-9

Short Answer

Step by step solution

Step 1 Given function is,

Step 2 The function H is in lower terms.

Step 3 In the given function degree of the numerator is greater than the degree of the denominator.

Step 4 Substitute Hx=x to determine whether the graph intersects at the asymptote y=x.

Step 5 Now plot a graph by using the known values.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Applied Mathematics

Calculus

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems $7 - 44$ , follow Steps $1$ through $7$ on page $228$ to analyze the graph of each function.
$H (x) = \frac{x^{3} - 1}{x^{2} - 9}$