/*! 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 52 Use a grapher to graph each of t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 52

Use a grapher to graph each of the following equations. On most graphers, equations must be solved for \(y\) before they can be entered. $$ x^{4}=y^{2}+x^{6} $$

Short Answer

Expert verified
To graph the equation, isolate y to get \[ y = \pm\sqrt{x^4 - x^6} \] and enter both the positive and negative square roots into the grapher.

Step by step solution

01

Isolate the Terms with y

Start by moving all the terms to one side of the equation to isolate terms with y. Rewrite the equation to \[ x^4 - x^6 = y^2 \]
02

Solve for y

To solve for y, take the square root of both sides. Remember to include both the positive and negative square roots. \[ y = \pm\sqrt{x^4 - x^6} \]
03

Enter the Equation into the Grapher

Use a graphing tool to input the two equations: \[ y = \sqrt{x^4 - x^6} \] and \[ y = -\sqrt{x^4 - x^6} \]. Observe the graphs of both equations.

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.

solving equations for y
When graphing an equation, it's often necessary to solve it for a particular variable. In this case, we solve for y. The original equation is: \(x^{4} = y^{2} + x^{6}\)Start by isolating terms that include y. You can move terms to one side like this:\[ x^4 - x^6 = y^2 \]Now that you have grouped the terms, solving for y involves taking the square root of both sides. Remember, this step gives you both a positive and a negative solution:\[ y = \pm\sqrt{x^4 - x^6} \]Understanding these steps is critical because any error here can lead to incorrect graphing results.
graphing with square roots
Square roots significantly impact how equations are graphed. When you encounter an equation with a square root, consider both the positive and negative roots. This is why we have two equations to graph for our example:\[ y = \sqrt{x^4 - x^6} \]and\[ y = -\sqrt{x^4 - x^6} \]Note that the range of x values must be chosen carefully. For example, the expression under the square root must be non-negative. Therefore, your valid x-values would be those making \(x^4 - x^6\) greater than or equal to zero. This often impacts the shape and domain of the graph greatly.
using graphing tools
Graphing tools are immensely helpful in visualizing complex equations. Most tools require the equation to be solved for y. Once you have your equations:\[ y = \sqrt{x^4 - x^6} \]and\[ y = -\sqrt{x^4 - x^6} \]You can enter these into the grapher. If you're using an online graphing calculator or software like Desmos, input each equation separately. The graph will show how the two curves mirror each other across the x-axis. Take note of:
  • The domain of the functions: which x-values result in real y-values.
  • The symmetry of the graphs.
  • If there are any asymptotes or x-intercepts.
Using these tools will allow you to check your algebraic work visually and deepen your understanding of function behaviors.

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 limit, if it exists. $$ \lim _{x \rightarrow-\infty} \frac{-6 x^{3}+7 x}{2 x^{2}-3 x-10} $$

Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. When no interval is specified, use the real line \((-\infty, \infty)\). $$ f(x)=\frac{1}{\sin x+\cos x} ; \quad(-\pi / 4,3 \pi / 4) $$

Find the absolute maximum and minimum values of the function, if they exist, over the indicated interval. When no interval is specified, use the real line \((-\infty, \infty)\). $$ f(x)=x^{4}-2 x^{2} $$

Look up the word implicit in a dictionary. Explain how that definition can be related to the concept of a function that is defined "implicitly."

Differentiate implicily to find \(d y / d x\). $$ \frac{x y}{x+y}=2 $$
See all solutions

Recommended explanations on Biology Textbooks

Cellular Energetics

Read Explanation

Reproduction

Read Explanation

Ecology

Read Explanation

Cell Communication

Read Explanation

Bioenergetics

Read Explanation

Microbiology

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.