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Chapter 3: Problem 61

For the graph of \(y=f(x)\), a. Identify the \(x\)-intercepts. b. Identify any vertical asymptotes. c. Identify the horizontal asymptote or slant asymptote if applicable. d. Identify the \(y\)-intercept. \(f(x)=\frac{(5 x-1)(x+3)}{x+2}\)

### Short Answer

Expert verified

x-intercepts: \( x = \frac{1}{5} \), \( x = -3 \).Vertical asymptote: \( x = -2 \).Slant asymptote: \( y = 5x - 2 \). y-intercept: \( (0, -\frac{3}{2}) \).

## Step by step solution

01

## Identify the x-intercepts

To find the x-intercepts, set the numerator of the function equal to zero and solve for x.\[ (5x - 1)(x + 3) = 0 \]This gives two solutions: \[ 5x - 1 = 0 \implies x = \frac{1}{5} \]and \[ x + 3 = 0 \implies x = -3 \] Thus, the x-intercepts are \( x = \frac{1}{5} \) and \( x = -3 \).

02

## Identify the vertical asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for x.\[ x + 2 = 0 \implies x = -2 \]Thus, there is a vertical asymptote at \( x = -2 \).

03

## Identify the horizontal asymptote or slant asymptote

To find the horizontal or slant asymptote, consider the degree of the numerator and the denominator. The numerator \( (5x-1)(x+3) \) is of degree 2, and the denominator \( x+2 \) is of degree 1. Since the degree of the numerator is greater than the degree of the denominator by 1, there is a slant asymptote.Perform polynomial long division of \( \frac{(5x^2 + 14x - 3)}{x+2} \) to find the slant asymptote.The result is:\[ 5x - 2 \]Thus, the slant asymptote is \( y = 5x - 2 \).

04

## Identify the y-intercept

To find the y-intercept, set \( x = 0 \) in the function and solve for \( y \).\[ f(0) = \frac{(5\cdot0 - 1)(0+3)}{0+2} = \frac{(0-1)(3)}{2} = \frac{-3}{2} \]Thus, the y-intercept is at \( (0, -\frac{3}{2}) \).

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### x-intercepts

The x-intercepts of a function are the points where the graph crosses the x-axis. This happens when the value of the function, f(x), is zero. To find them, set the numerator of the fraction form of the function f(x) to zero and solve for x. In this exercise, the function is given by: \[ f(x) = \frac{(5x-1)(x+3)}{x+2} \] Set the numerator equal to zero: \[ (5x - 1)(x + 3) = 0 \] Solving this equation, we first get: \[ 5x - 1 = 0 \implies x = \frac{1}{5} \] And then: \[ x + 3 = 0 \implies x = -3 \] Thus, the x-intercepts are \( x = \frac{1}{5} \) and \( x = -3 \).

###### vertical asymptotes

Vertical asymptotes occur where the function becomes undefined – typically, where the denominator of the fractional part of the function is zero. The graph approaches these lines but never actually touches or crosses them. To find vertical asymptotes: Set the denominator of \( f(x) = \frac{(5x-1)(x+3)}{x+2} \) to zero: \[ x + 2 = 0 \implies x = -2 \] Thus, there is a vertical asymptote at \( x = -2 \).

###### horizontal asymptotes

Horizontal asymptotes are lines that the graph of the function approaches as x tends towards infinity or negative infinity. For rational functions, they depend on the degrees of the numerator and the denominator.

- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
- If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote, but there may be a slant asymptote.

###### slant asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. The asymptote is a linear equation, and you find it by performing polynomial long division. For the given function: \[ f(x) = \frac{(5x-1)(x+3)}{x+2} \] First, simplify the numerator: \[(5x-1)(x+3) = 5x^2 + 14x - 3 \] Then, divide \( 5x^2 + 14x - 3 \) by \( x + 2 \). The result of the division provides the equation of the slant asymptote: \[ 5x - 2 \] Thus, the slant asymptote is y = 5x - 2.

###### y-intercepts

The y-intercept of a function is where the graph crosses the y-axis. This occurs when x = 0. To find it, substitute x = 0 into the function and solve for y. For the function: \[ f(x) = \frac{(5x-1)(x+3)}{x+2} \] Set x = 0: \[ f(0) = \frac{(5 \cdot 0 - 1)(0 + 3)}{0 + 2} = \frac{(0 - 1) \cdot 3}{2} = \frac{-3}{2} \] Thus, the y-intercept is at (0, -\( \frac{3}{2} \)).

###### polynomial long division

Polynomial long division is used to divide two polynomials when finding slant asymptotes. It's similar to numerical long division but with variables.

- Divide the first term of the numerator by the first term of the denominator.
- Multiply the entire denominator by that result and subtract from the original polynomial.
- Repeat with the resulting polynomial.

For the function \( f(x) = \frac{5x^2 + 14x - 3}{x + 2} \), divide step-by-step to get the slant asymptote y = 5x - 2.

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