/*! 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 58 Find any intercepts of the graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 58

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=\frac{x^{2}-10}{x^{2}+10}\)

Short Answer

Expert verified
X-intercepts: \((\sqrt{10}, 0), (-\sqrt{10}, 0)\); Y-intercept: \((0, -1)\). Symmetric w.r.t. y-axis.

Step by step solution

01

Find the x-intercepts

To find the x-intercepts, set the equation equal to zero and solve for \(x\): \(y = \frac{x^2 - 10}{x^2 + 10} = 0\). This equation is equal to zero when the numerator is equal to zero, provided the denominator is not zero. Therefore, \(x^2 - 10 = 0\). Solving for \(x\), we get \(x^2 = 10\) or \(x = \pm \sqrt{10}\). Thus, the x-intercepts are at \((\sqrt{10}, 0)\) and \((-\sqrt{10}, 0)\).
02

Check for y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\): \(y = \frac{(0)^2 - 10}{(0)^2 + 10} = \frac{-10}{10} = -1\). Thus, the y-intercept is at \((0, -1)\).
03

Test for symmetry with respect to the x-axis

A graph is symmetric with respect to the x-axis if replacing \(y\) with \(-y\) in the equation yields an equivalent equation. Testing: \(-y = \frac{x^2 - 10}{x^2 + 10}\). Solving for \(y\), we get \(y = -\frac{x^2 - 10}{x^2 + 10}\), which is not equivalent to the original equation. Therefore, the graph is not symmetric with respect to the x-axis.
04

Test for symmetry with respect to the y-axis

A graph is symmetric with respect to the y-axis if replacing \(x\) with \(-x\) in the equation yields an equivalent equation. Testing: \(y = \frac{(-x)^2 - 10}{(-x)^2 + 10} = \frac{x^2 - 10}{x^2 + 10}\). Since the equation is unchanged, the graph is symmetric with respect to the y-axis.
05

Test for symmetry with respect to the origin

A graph is symmetric with respect to the origin if replacing \(x\) with \(-x\) and \(y\) with \(-y\) yields an equivalent equation. Testing: \(-y = \frac{(-x)^2 - 10}{(-x)^2 + 10} = \frac{x^2 - 10}{x^2 + 10}\). Solving for \(y\), \(y = -\frac{x^2 - 10}{x^2 + 10}\), which is not equivalent to the original equation. Thus, the graph is not 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!

Key Concepts

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

Understanding x-intercepts
The concept of an x-intercept is quite straightforward. It's where the graph of an equation crosses the x-axis. In mathematical terms, this happens when the value of y is zero. Finding the x-intercepts of an equation involves setting the y to zero and solving for x.

For the equation given, the task was to solve \( y = \frac{x^2 - 10}{x^2 + 10} = 0 \). By making the numerator zero (as a fraction is zero only when its numerator is zero), we got \(x^2 - 10 = 0\). Solving this gives \(x = \pm \sqrt{10}\).

So, the graph crosses the x-axis at two points: \( (\sqrt{10}, 0) \) and \( (-\sqrt{10}, 0) \). Always remember:
  • X-intercepts happen when \( y = 0 \).
  • Set the numerator to zero when dealing with rational functions to find x-intercepts.
Exploring y-intercepts
Y-intercepts are where the graph meets the y-axis. Finding them is simpler than x-intercepts. All you need to do is set \(x = 0\) in the equation and solve for y. It tells you where the graph crosses the y-axis.

For the given equation, substituting \(x = 0\) resulted in \( y = \frac{(0)^2 - 10}{(0)^2 + 10} = \frac{-10}{10} = -1 \). It means the y-intercept is at the point \( (0, -1) \).

To sum up:
  • Set \( x = 0 \) to find the y-intercept.
  • The y-intercept is where the graph intersects the y-axis.
Delving into symmetry
Symmetry in graphs can help you understand the nature of a function's graph, simplifying how you think about its shape and behavior. There are three main types of symmetry you might consider:
  • Symmetry with respect to the x-axis: This happens when replacing \(y\) with \(-y\) gives the same equation. Our example didn't have this symmetry, as evidenced when we tested it.
  • Symmetry with respect to the y-axis: Occurs if by replacing \(x\) with \(-x\), you get an equivalent equation. For the equation in question, this symmetry existed because replacing \(x\) with \(-x\) did not change the equation: \( y = \frac{x^2 - 10}{x^2 + 10} \).
  • Origin symmetry: This symmetry appears when swapping both \(x\) with \(-x\) and \(y\) with \(-y\) results in the same equation. In our case, the graph wasn't symmetric with the origin since that substitution altered the equation.
Understanding these can help predict how the graph of an equation behaves without plotting it explicitly. Remember:
  • Graphs symmetric about the y-axis are mirror images across the y-axis.
  • Graphs symmetric about the x-axis are mirror images across the x-axis, and for symmetric about the origin graphs, rotating 180 degrees results in the same graph.

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

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{3}+3 x^{2}+3 x+1}{x^{4}+x^{3}+x+1}\) (b) \(\lim _{x \rightarrow-1} \frac{x^{3}+3 x^{2}+3 x+1}{x^{4}+x^{3}+x+1}\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=|x-9|\)

Use addition of algebraic fractions to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{x^{2}+3 x-1}{x}+\frac{1}{x}\) (b) \(\lim _{x \rightarrow 0}\left[\frac{x^{2}+3 x-1}{x}+\frac{1}{x}\right]\)

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{5-5(h+1)^{2}}{h}\) (b) \(\lim _{h \rightarrow 0} \frac{5-5(h+1)^{2}}{h}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.