Explanation Given information: The rational function is s ( x ) = 3 x 2 + 6 x 2 − 2 x − 3 . Calculation: The rational function s ( x ) can be written in its factor form as, s ( x ) = 3 ( x 2 + 2 ) ( x − 3 ) ( x + 1 ) ...... (1) The x- intercept of the rational function is the point such that the numerator of the function is zero. The numerator of the rational function s ( x ) is 3 x 2 + 6 . Here, the x - intercept of the rational function is greater than zero. Therefore, the x -intercept of the rational function does not exist. The y -intercept of the rational function are the points such that x is equals to zero. Substitute 0 for x in the equation (1) to obtain they -intercept of the rational function as, s ( 0 ) = 3 ( 0 2 + 2 ) ( 0 − 3 ) ( 0 + 1 ) = 6 − 3 = − 2 The value of s ( 0 ) is equal to − 2 . Therefore, the y -intercept of the rational function is − 2 . The vertical asymptote of the rational function is a line x = a such that y approaches to ± ∞ if x approaches to a. At the point of vertical asymptote the denominator of the rational function is zero. The denominator of the function s ( x ) from the equation (1) is ( x − 3 ) ( x + 1 ) . Now, equate the denominator of the rational function s ( x ) equals to zero to obtain the vertical asymptote of the rational function as, ( x − 3 ) ( x + 1 ) = 0 x − 3 = 0 or x + 1 = 0 x = 3 or x = − 1 Therefore, the vertical asymptote of the function is x = − 1 and x = 0 . The horizontal asymptote is the value of y such that x approaches to + ∞ or − ∞ . The horizontal asymptote of the rational function is y = b . Here, b is the ratio of coefficient of x that has highest degree in numerator and denominator of rational function s ( x ) as, y = ( 3 ) ⋅ ( 1 ) ( 1 ) ⋅ ( 1 ) y = 3 Therefore, the horizontal asymptote of the function is y = 3 . Graph: Obtain the value of y for different values of x to plot the graph of rational function s ( x ) . Substitute 0 for xin equation (1) to find the value of s ( 0 ) as, s ( 0 ) = 3 ( 0 2 + 2 ) ( 0 − 3 ) ( 0 + 1 ) = 6 − 3 = − 2 Substitute 1 for xin equation (1) to find the value of s ( 1 ) as follows, s ( 1 ) = 3 ( 1 2 + 2 ) ( 1 − 3 ) ( 1 + 1 ) = 9 − 4 = − 9 4 Substitute − 2 for xin equation (1) to find the value of s ( − 2 ) as follows, s ( − 2 ) = 3 ( ( − 2 ) 2 + 2 ) ( − 2 − 3 ) ( − 2 + 1 ) = 18 5 Similarly, obtain the other coordinate points that lie on the curve and list all the points in form of table as shown in Table 1

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 3.6, Problem 56E

To determine

To find:

The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.6, Problem 55E

Chapter 3.6, Problem 57E

Students have asked these similar questions

Economists use what is called a Leffer curve to predict the government revenue for tax rates from 0% to 100%. Economists agree that the end points of the curve generate 0 revenue, 10x(100-x) , where R is revenue in millions at a tax but disagree on the tax rate that produces the maximum revenue. Suppose an economist produces this rational function R(x) = 75+ x rate of x percent. Use a graphing calculator to graph the function. What tax rate produces the maximum revenue? What is the maximum revenue? OA. 39.6%; $209 million OB. 35.8%; $209 million OC. 37.5%; $210 million OD. 34.9%; $207 million C

need help answeting this question. Thanks!

Use the given graph to create the equation for the rational function. The function is written in factored form to help you see how the given information shapes our equation The numerator is: 10 Check (x-3) |) The denominator is: (x+ 3 )(x-4 )

Chapter 3 Solutions

College Algebra

Ch. 3.1 - To put the quadratic function f(x)=ax2+bx+c in...Ch. 3.1 - The quadratic function f(x)=a(xh)2+k is in...Ch. 3.1 - The graph of f(x)=3(x2)26 is a parabola that...Ch. 3.1 - Prob. 4E Ch. 3.1 - Graphs of Quadratic Functions The graph of a...Ch. 3.1 - Prob. 6E Ch. 3.1 - Graphs of Quadratic Functions The graph of a...Ch. 3.1 - Prob. 8E Ch. 3.1 - Graphing Quadratic Functions A quadratic function...Ch. 3.1 - Graphing Quadratic Functions A quadratic function...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The intercepts and asymptotes of the rational function s ( x ) . Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Solution Summary: The author explains the rational function s(x) and its intercepts and asymptotes.

Videos

To find:

The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Want to see the full answer?

Chapter 3 Solutions

The intercepts and asymptotes of the rational function s ( x ) . Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Solution Summary: The author explains the rational function s(x) and its intercepts and asymptotes.

Videos

To find: The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Want to see the full answer?

Chapter 3 Solutions

To find:

The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.