/*! 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 36 Without graphing, determine whet... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 36

Without graphing, determine whether each equation has a graph that is symmetric with respect to the \(x\) -axis, the \(y\) -axis, the origin, or none of these. $$y=2 x^{4}-3$$

Short Answer

Expert verified
The equation is symmetric with respect to the y-axis.

Step by step solution

01

Identify Symmetry with the x-axis

For symmetry with respect to the x-axis, replace y with -y in the equation. Start with the original equation: y = 2x^4 - 3Replace y with -y: -y = 2x^4 - 3Solve for y:y = -2x^4 + 3The resulting equation is not the same as the original equation, therefore it is not symmetric with respect to the x-axis.
02

Identify Symmetry with the y-axis

For symmetry with respect to the y-axis, replace x with -x in the equation. Start with the original equation: y = 2x^4 - 3Replace x with -x:y = 2(-x)^4 - 3Simplify:y = 2x^4 - 3The resulting equation is the same as the original, therefore it is symmetric with respect to the y-axis.
03

Identify Symmetry with the Origin

For symmetry with respect to the origin, replace both x with -x and y with -y in the equation. Start with the original equation: y = 2x^4 - 3Replace x with -x and y with -y: -y = 2(-x)^4 - 3Simplify:y = -2x^4 + 3The resulting equation is not the same as the original equation, therefore it 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.

x-axis symmetry
To determine if a graph has symmetry with respect to the x-axis, you need to replace every instance of y in the equation with -y and then simplify.

If the new equation looks the same as the original equation, then the graph has x-axis symmetry. For instance, consider the equation given:

y = 2x^{4} - 3
Replace y with -y: -y = 2x^{4} - 3
Now solve for y by multiplying both sides by -1:
y = -2x^{4} + 3
The original equation was y = 2x^{4} - 3, which is not the same as y = -2x^{4} + 3, so this graph does not have x-axis symmetry.

Simplified Explanation:

  • Replace y with -y.
  • Simplify the equation.
  • If it matches the original, it has x-axis symmetry; if not, it does not.
y-axis symmetry
Checking for y-axis symmetry involves replacing every x in the equation with -x. If the equation still looks the same after this change, the graph has y-axis symmetry. Starting with our given equation:

y = 2x^{4} - 3
Replace x with -x: y = 2(-x)^{4} - 3
Simplify using the fact that (-x)^{4} = x^{4} regardless of x being positive or negative:
y = 2x^{4} - 3
Since the modified equation matches the original, the graph has y-axis symmetry.

Simplified Explanation:
  • Replace x with -x.
  • Simplify the equation.
  • If it matches the original, it has y-axis symmetry; if not, it does not.
origin symmetry
To test for origin symmetry, change both x to -x and y to -y in the equation. If the equation remains unchanged, then it has origin symmetry. Let's do this with our given equation:

y = 2x^{4} - 3
Replace both x with -x and y with -y: -y = 2(-x)^{4} - 3
Simplify using the fact that (-x)^{4} = x^{4}:
-y = 2x^{4} - 3
Now solve for y by multiplying both sides by -1:
y = -2x^{4} + 3
The original equation was y = 2x^{4} - 3, so this graph does not have origin symmetry, because the simplified equation does not match the original.

Simplified Explanation:
  • Replace both x with -x and y with -y.
  • Simplify the equation.
  • If it matches the original, it has origin symmetry; if not, it does not.

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

Find the center-radius form of the circle described or graphed. a circle having a diameter with endpoints \((5,4)\) and \((-3,-2)\)

Find the equation of a circle with center at \((-4,3),\) passing through the point \((5,8) .\) Write it in center-radius form.

Determine whether the three points are collinear by using slopes. $$(0,9),(-3,-7),(2,19)$$

Let \(f(x)=2 x-3\) and \(g(x)=-x+3 .\) Find each function value. $$(f \circ g)(-2)$$

Determine whether the three points are collinear by using slopes. $$(-1,4),(-2,-1),(1,14)$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.