Question

Graph each function using technology and identify any asymptotes and intercepts. a) \(y=\frac{2 x+1}{x-4}\) b) \(y=\frac{3 x-2}{x+1}\) c) \(y=\frac{-4 x+3}{x+2}\) d) \(y=\frac{2-6 x}{x-5}\)

Short answer

For each function: (a) Asymptotes: x=4, y=2; Intercepts: x=-1/2, y=-1/4(b) Asymptotes: x=-1, y=3; Intercepts: x=2/3, y=-2(c) Asymptotes: x=-2, y=-4; Intercepts: x=3/4, y=3/2(d) Asymptotes: x=5, y=-6; Intercepts: x=1/3, y=-2/5

Step-by-step solution

Graph the Function

Use graphing technology (a graphing calculator or software) to plot each function. Observe the shape and features of each graph carefully.

Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function equals zero, causing the function to be undefined. Set the denominator equal to zero and solve for x.

Determine Horizontal Asymptotes

For rational functions, compare 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 degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.

Find the Intercepts

To find x-intercepts, set the numerator equal to zero and solve for x. For y-intercepts, evaluate the function at x=0.

Analyze Each Function Separately

Let's proceed to analyze each given function.

Function (a) Analysis

For the function \(y=\frac{2x+1}{x-4}\): 1. Vertical asymptote: Set x-4=0, thus x=4. 2. Horizontal asymptote: Degrees are equal, so it is y=\frac{2}{1}=2. 3. X-intercept: Set 2x+1=0, thus x=-\frac{1}{2}. 4. Y-intercept: Evaluate at x=0, thus y=\frac{2(0)+1}{0-4}=-\frac{1}{4}.

Function (b) Analysis

For the function \(y=\frac{3x-2}{x+1}\): 1. Vertical asymptote: Set x+1=0, thus x=-1. 2. Horizontal asymptote: Degrees are equal, so it is y=\frac{3}{1}=3. 3. X-intercept: Set 3x-2=0, thus x=\frac{2}{3}. 4. Y-intercept: Evaluate at x=0, thus y=\frac{3(0)-2}{0+1}=-2.

Function (c) Analysis

For the function \(y=\frac{-4x+3}{x+2}\): 1. Vertical asymptote: Set x+2=0, thus x=-2. 2. Horizontal asymptote: Degrees are equal, so it is y=\frac{-4}{1}=-4. 3. X-intercept: Set -4x+3=0, thus x=\frac{3}{4}. 4. Y-intercept: Evaluate at x=0, thus y=\frac{-4(0)+3}{0+2}=\frac{3}{2}.

Function (d) Analysis

For the function \(y=\frac{2-6x}{x-5}\): 1. Vertical asymptote: Set x-5=0, thus x=5. 2. Horizontal asymptote: Degrees are equal, so it is y=\frac{-6}{1}=-6. 3. X-intercept: Set 2-6x=0, thus x=\frac{1}{3}. 4. Y-intercept: Evaluate at x=0, thus y=\frac{2-6(0)}{0-5}=-\frac{2}{5}.

Key concepts

Vertical Asymptotes

Vertical asymptotes occur where the function becomes undefined, which happens when the denominator is zero. For a rational function like \( y = \frac{p(x)}{q(x)} \), setting \( q(x) = 0 \) and solving for \( x \) will give you the vertical asymptotes. For example, for the function \( y = \frac{2x+1}{x-4} \), setting \( x-4 = 0 \) gives \( x = 4 \). Thus, \( x = 4 \) is a vertical asymptote. Practically, vertical asymptotes are lines where the graph shoots off to infinity or negative infinity.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of the function as \( x \) approaches infinity or negative infinity. They're determined by comparing the degrees of the numerator and denominator of the rational function. For \( y = \frac{2x + 1}{x - 4} \), both the numerator and denominator have a degree of 1. Here, the horizontal asymptote is the ratio of the leading coefficients, \( y = \frac{2}{1} = 2 \). If the numerator's degree is less than the denominator's, the horizontal asymptote is \( y = 0 \).

X-Intercepts

X-intercepts are points where the graph crosses the x-axis, which occur when \( y = 0 \). To find them, set the numerator equal to zero and solve for \( x \). For \( y = \frac{3x-2}{x+1} \), setting \( 3x - 2 = 0 \) and solving gives \( x = \frac{2}{3} \). This means the graph crosses the x-axis at \( x = \frac{2}{3} \). Understanding x-intercepts is crucial as they give insights where the function value is zero.

Y-Intercepts

Y-intercepts are points where the graph crosses the y-axis, which occur when \( x = 0 \). To find them, evaluate the function at \( x = 0 \). For \( y = \frac{-4x+3}{x+2} \), substituting \( x = 0 \) gives \( y = \frac{3}{2} \). Therefore, the graph crosses the y-axis at \( y = \frac{3}{2} \). Y-intercepts illustrate the initial value or starting point of the function where x is zero.

Rational Function Analysis

Analyzing rational functions involves understanding their properties and behaviors, such as vertical and horizontal asymptotes, x-intercepts, and y-intercepts. For example, let's examine \( y = \frac{2-6x}{x-5} \):

• Vertical Asymptote: Set \( x-5=0 \), so \( x=5 \).
• Horizontal Asymptote: Degrees are equal, take ratio of leading coefficients so \( y= -6 \).
• X-intercept: Set \(2 - 6x = 0 \), solving for \( x = \frac{1}{3} \).
• Y-intercept: Substitute \( x = 0 \), result is \( y = -\frac{2}{5} \).

This structured analysis helps breaking down complex functions into understandable parts.