Problem 40

Question

For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph. $$ q(x)=\frac{x-5}{3 x-1} $$

Step-by-Step Solution

Verified

Answer

Intercepts: (5,0) and (0,5); Asymptotes: x = 1/3 and y = 1/3.

1Step 1: Find the Horizontal Intercept

To find the horizontal intercept of the function, set the numerator equal to zero and solve for x: \ \[ x-5 = 0 \] \ Solution gives us the horizontal intercept: \ \[ x = 5 \].

2Step 2: Find the Vertical Intercept

To find the vertical intercept of the function, set $ x = 0 $ and solve for $ q(0) $: \ \[ q(0) = \frac{0-5}{3(0)-1} = \frac{-5}{-1} = 5 \] \ The vertical intercept is $ y = 5 $.

3Step 3: Identify the Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (provided these points are not canceled by zeros in the numerator). Set the denominator to zero: \ \[ 3x-1 = 0 \] \ Solve for $ x $: \ \[ x = \frac{1}{3} \] \ This is the vertical asymptote.

4Step 4: Determine the Horizontal Asymptote

For rational functions, the horizontal asymptote depends on the degrees of the polynomials. Both the numerator and denominator have the highest degree of one. The horizontal asymptote is given by the ratio of the leading coefficients: \ \[ y = \frac{1}{3} \].

5Step 5: Sketch the Graph

Plot the intercepts and asymptotes on a coordinate plane. Draw lines approaching the asymptotes, taking care to adjust near the intercepts: $ x = 5 $ and $ y = 5 $. Keep in mind the curve approaches the lines $ x = \frac{1}{3} $ and $ y = \frac{1}{3} $ infinitely.

Key Concepts

Horizontal interceptsVertical interceptsAsymptotesGraph sketching

Horizontal intercepts

Finding horizontal intercepts is an important skill when working with rational functions. To find the horizontal intercept of a rational function like $ q(x) = \frac{x-5}{3x-1} $, you set the numerator equal to zero. This means solving $ x - 5 = 0 $, which gives us $ x = 5 $.

Horizontal intercept: where the graph crosses the x-axis.
In our case, the horizontal intercept is $ (5, 0) $.

Horizontal intercepts tell us where the function touches or crosses the x-axis. For rational functions, it is only possible if the numerator becomes zero while the denominator is non-zero at the same x-value.

Vertical intercepts

Vertical intercepts occur where the graph crosses the y-axis. This happens when $ x = 0 $. To find the vertical intercept for the function $ q(x) = \frac{x-5}{3x-1} $, just substitute 0 into the equation: \[ q(0) = \frac{0-5}{3(0)-1} = 5 \] This tells us that the vertical intercept is at $ y = 5 $, so the point $ (0, 5) $ is on the graph.

Vertical intercept: where the function crosses the y-axis.
For $ q(x) $, it is $ (0, 5) $.

This intercept is useful because it provides a starting point for sketching the graph of the rational function. It is an easy x-value to compute and provides immediate visual information.

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. They are crucial in identifying the behavior of rational functions at the extremes and can be classified into vertical and horizontal asymptotes. **Vertical Asymptotes** These occur where the denominator of the rational function becomes zero (and is not canceled by the numerator). For $ q(x) = \frac{x-5}{3x-1} $, set $ 3x-1 = 0 $ and solve to find $ x = \frac{1}{3} $. Thus, the vertical asymptote is at $ x = \frac{1}{3} $.
**Horizontal Asymptotes** These are determined by comparing the degrees of the numerator and denominator. In $ q(x) $, both have a degree of one, so we find the horizontal asymptote by dividing the leading coefficients: \[ y = \frac{1}{3} \]

Vertical asymptotes show values where the function turns infinite.
Horizontal asymptotes show end behavior as $ x \to \pm \infty $.

Asymptotes provide insight into the long-term behavior of the function and are vital for graph sketching.

Graph sketching

Graph sketching of rational functions involves incorporating all previously discovered intercepts and asymptotes into a visual representation. Start by plotting the horizontal intercept at $ (5, 0) $ and the vertical intercept at $ (0, 5) $.

Draw the vertical asymptote line at $ x = \frac{1}{3} $.
Draw the horizontal asymptote line at $ y = \frac{1}{3} $.

To sketch the graph, think of how the curve approaches the asymptotes at extreme values. The graph will bend closer yet never touch these asymptotic lines. The crossing of the intercepts helps to shape where the graph is likely to bend or change directions near those points. With rational functions, it is helpful to also approach the problem with intuition and understand the general form the graph might take, often resembling a hyperbola depending on the specific function form.

Problem 40

Other exercises in this chapter

Problem 40

For the following exercises, use the given information to find the unknown value. $y$ varies jointly as the square of $x$ and of $z$ and inversely as the

View solution

Problem 40

For the following exercises, find all complex solutions (real and non-real). $x^{3}+x^{2}+x+1=0$

View solution

Problem 40

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=\frac{1}{x^{2}}, x \geq 0 $$

View solution

Problem 40

For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization. $

View solution