Chapter 4: Q. 36 (page 234)

In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$

Short Answer

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The graph is

Step by step solution

Given information

Given function $R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$

Factoring the numerator and denominator

Factoring, we get

$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15} R (x) = \frac{8 x^{2} + 30 x + 4 x + 15}{2 x^{2} - 6 x + 5 x - 15} R (x) = \frac{2 x (4 x + 15) + 1 (4 x + 15)}{2 x (x - 3) + 5 (x - 3)} R (x) = \frac{(2 x + 1) (4 x + 15)}{(2 x + 5) (x - 3)}$

Calculating the domain

Setting the denominator to zero, we get

$(2 x + 5) (x - 3) = 0 x = - \frac{5}{2}, 3$

The domain is all real numbers except $- \frac{5}{2}, 3$

Calculating the x intercept

Setting numerator to zero, we get

$(2 x + 1) (4 x + 15) = 0 x = - \frac{1}{2}, \frac{- 4}{15}$

The $y$ intercept is $R (0) = - 1$ i.e. $(0, - 1)$

Calculating the vertical asymptote

The zeroes of denominator is $x = - \frac{5}{2}, 3$ so the vertical asymptote is $x = - \frac{5}{2}, 3$

Calculating the horizontal asymptote

The degree of numerator is equal to denominator so the horizontal asymptote is

$y = \frac{a}{b} y = \frac{1}{1} y = 1$

$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15} = 4 8 x^{2} + 26 x + 15 = 4 (2 x^{2} - x - 15) 8 x^{2} + 26 x + 15 = 8 x^{2} - 8 x - 60 x = - \frac{15}{7}$

The intersection point is $(- \frac{15}{7}, 4)$

Sketching the graph

The graph is

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In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.
$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculating the domain

Calculating the x intercept

Calculating the vertical asymptote

Calculating the horizontal asymptote

Sketching the graph

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In Problems 7-44, follow Steps 1 through 7 on page 228 to analyze the graph of each function.R(x)=8x2+26x+152x2-x-15

Short Answer

Step by step solution

Given information

Factoring the numerator and denominator

Calculating the domain

Calculating the x intercept

Calculating the vertical asymptote

Calculating the horizontal asymptote

Sketching the graph

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Geometry

Pure Maths

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.
$R (x) = \frac{8 x^{2} + 26 x + 15}{2 x^{2} - x - 15}$