/*! 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 17 Determine whether the graph of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 17

Determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, origin, or none of these. $$ x^{2}+y^{2}=3 $$

Short Answer

Expert verified
The graph is symmetric with respect to the x-axis, y-axis, and origin.

Step by step solution

01

Understand the problem

Determine if the graph of the equation is symmetric with respect to the x-axis, y-axis, origin, or none of these. The equation given is \(x^2 + y^2 = 3\).
02

Test for symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, replace \(y\) with \(-y\) in the equation. Original equation: \(x^2 + y^2 = 3\).After replacing \(y\) with \(-y\): \(x^2 + (-y)^2 = 3\) which simplifies to \(x^2 + y^2 = 3\).Since the equation remains unchanged, it is symmetric with respect to the x-axis.
03

Test for symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace \(x\) with \(-x\) in the equation. Original equation: \(x^2 + y^2 = 3\).After replacing \(x\) with \(-x\): \((-x)^2 + y^2 = 3\) which simplifies to \(x^2 + y^2 = 3\).Since the equation remains unchanged, it is symmetric with respect to the y-axis.
04

Test for symmetry with respect to the origin

To test for symmetry with respect to the origin, replace \((x, y)\) with \((-x, -y)\) in the equation.Original equation: \(x^2 + y^2 = 3\).After replacing \(x\) with \(-x\) and \(y\) with \(-y\): \((-x)^2 + (-y)^2 = 3\) which simplifies to \(x^2 + y^2 = 3\).Since the equation remains unchanged, it is 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.

x-axis symmetry
To determine if a graph is symmetric with respect to the x-axis, you need to test if flipping the graph over the x-axis yields the same equation. This is done by replacing every instance of y in the equation with -y.
For example, with the equation \( x^2 + y^2 = 3 \), you substitute y with -y:
  • Original equation: \( x^2 + y^2 = 3 \)
  • Replace y with -y: \( x^2 + (-y)^2 = 3 \)
  • Simplify: \( x^2 + y^2 = 3 \)
Since the equation remains unchanged, the graph of \( x^2 + y^2 = 3 \) is symmetric with respect to the x-axis.
This means if you fold the graph over the x-axis, both halves will match perfectly.
y-axis symmetry
Checking for y-axis symmetry involves a similar process but here, you replace x with -x instead.
Consider the same equation,\( x^2 + y^2 = 3 \). Substitute x with -x:
  • Original equation: \( x^2 + y^2 = 3 \)
  • Replace x with -x: \( (-x)^2 + y^2 = 3 \)
  • Simplify: \( x^2 + y^2 = 3 \)
Since the equation remains the same, the graph is symmetric with respect to the y-axis.
If you fold the graph over the y-axis, both sides will align perfectly. This property is crucial in understanding the overall structure of the graph.
origin symmetry
To test for symmetry with respect to the origin, replace both x and y with their negative counterparts, -x and -y.
Let's revisit our equation \( x^2 + y^2 = 3 \). Replace both x and y with -x and -y:
  • Original equation: \( x^2 + y^2 = 3 \)
  • Replace x with -x and y with -y: \( (-x)^2 + (-y)^2 = 3 \)
  • Simplify: \( x^2 + y^2 = 3 \)
Since the equation doesn't change, it shows that the graph is symmetric about the origin.
This symmetry means that if you rotate the graph 180 degrees around the origin, it will look the same as it did before the rotation.
equation transformation
Equation transformation is a method used to test for various symmetries in a graph. It involves substituting variables and simplifying the equation.
For x-axis symmetry, replace y with -y. For y-axis symmetry, replace x with -x. And for origin symmetry, replace both x and y with -x and -y respectively.
Let's summarize our transformation checks using \( x^2 + y^2 = 3 \):
  • x-axis symmetry: \( x^2 + (-y)^2 = 3 \)
  • y-axis symmetry: \( (-x)^2 + y^2 = 3 \)
  • origin symmetry: \( (-x)^2 + (-y)^2 = 3 \)
Each transformation leaves our equation unchanged, confirming that the graph exhibits x-axis, y-axis, and origin symmetry.
Understanding these transformations helps in analyzing and predicting the behavior of different graphs.

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

Given \(f(x)=4 \sqrt{x}\) a. Find the difference quotient (do not simplify). b. Evaluate the difference quotient for \(x=1\), and the following values of \(h: h=1, h=0.1, h=0.01,\) and \(h=0.001\). Round to 4 decimal places. c. What value does the difference quotient seem to be approaching as \(h\) gets close to \(0 ?\)

Graph the function. $$ g(x)=\left\\{\begin{aligned} x+2 & \text { for } x<-1 \\ -x+2 & \text { for } x \geq-1 \end{aligned}\right. $$

Graph the function. $$ f(x)=\left\\{\begin{array}{ll} |x| & \text { for } x<2 \\ -x+4 & \text { for } x \geq 2 \end{array}\right. $$

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$\left(\frac{r}{q}\right)(x)$$

Refer to the functions \(f, g,\) and \(h\) and evaluate the given functions. \(f(x)=2 x+1 \quad g(x)=x^{2} \quad h(x)=\sqrt[3]{x}\) $$(g \circ h \circ f)(x)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.