/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 Find \(a\) and \(b\) if the grap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 89

Find \(a\) and \(b\) if the graph of \(y=a x^{2}+b x^{3}\) is symmetric with respect to (a) the \(y\) -axis and (b) the origin. (There are many correct answers.)

Short Answer

Expert verified
For the function to be symmetric with respect to the y-axis, we need \(b = 0\), therefore any pair \((a, 0)\) where \(a\) is any real number is a solution. For the function to be symmetric with respect to the origin, we need \(a = 0\), hence any pair \((0, b)\), where \(b\) is any real number is a solution.

Step by step solution

01

Determine Symmetry with Respect to the Y-Axis

To find the values of \(a\) and \(b\) for which the function \(y = ax^2+bx^3\) is symmetric about the y-axis, use the constraint that an even function must satisfy \(f(-x)=f(x)\). Substituting this into the equation yields \(a(-x)^2 + b(-x)^3 = ax^2 + bx^3\). This simplifies to \(ax^2 - bx^3 = ax^2 + bx^3\). The only way for this equation to hold for all \(x\) is if \(b = 0\). Therefore, all pairs \((a, 0)\) will make the function symmetric with respect to the y-axis.
02

Determine Symmetry with Respect to the Origin

Next, to find the values of \(a\) and \(b\) for which the graph is symmetric about the origin, use the constraint for odd functions, specifically \(f(-x) = -f(x)\). Substituting this into the equation results in \(a(-x)^2 + b(-x)^3 = -(ax^2 + bx^3)\). Simplifying this expression gives \(ax^2 - bx^3 = -ax^2 - bx^3\), which only holds for all \(x\) if \(a = 0\). So, the pairs \((0, b)\) will make the function symmetric with respect to the origin.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\frac{x+1}{x-2} $$

Write a sentence using the variation terminology of this section to describe the formula. Surface area of a sphere: \(S=4 \pi r^{2}\)

Determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f \mathrm{~h}\) a \(y\) -intercept, then the \(y\) -intercept of \(f\) is an \(x\) -intercep of \(f^{-1}\)

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \hline \end{array} $$

The winning times (in minutes) in the women's 400-meter freestyle swimming event in the Olympics from 1948 through 2008 are given by the following ordered pairs. $$\begin{array}{lll} (1948,5.30) & (1972,4.32) & (1996,4.12) \\ (1952,5.20) & (1976,4.16) & (2000,4.10) \\ (1956,4.91) & (1980,4.15) & (2004,4.09) \\ (1960,4.84) & (1984,4.12) & (2008,4.05) \\ (1964,4.72) & (1988,4.06) & \\ (1968,4.53) & (1992,4.12) & \end{array}$$ A linear model that approximates the data is \(y=-0.020 t+5.00,\) where \(y\) represents the winning time (in minutes) and \(t=0\) represents \(1950 .\) Plot the actual data and the model on the same set of coordinate axes. How closely does the model represent the data? Does it appear that another type of model may be a better fit? Explain. (Source: International Olympic Committee)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.