/*! 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 19 Find the \(x\) - and \(y\) -inte... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ x^{2} y-x^{2}+4 y=0 $$

Short Answer

Expert verified
The x-intercept and y-intercept of the equation is at the point (0, 0).

Step by step solution

01

Find the x-intercepts

Set \(y=0\) in the equation to find the x-intercepts. Substituting \(y=0\) gives \(x^{2}(0) - x^{2} + 4(0) = 0\), which simplifies to \(-x^{2} = 0\). Solving for \(x\) gives us \(x=0\). So, the x-intercept is at the point (0, 0).
02

Find the y-intercepts

Set \(x=0\) in the equation to find the y-intercept. Substituting \(x=0\) gives \(0^{2}y - 0^{2} + 4y = 0\), which simplifies to \(4y=0\). Solving for \(y\) gives \(y=0\). So, the y-intercept is at the point (0, 0).

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-intercepts
To find the x-intercepts of an equation, we need to determine where the graph of the equation crosses the x-axis. On the x-axis, the value of y is always zero. Thus, to find x-intercepts, we substitute \(y = 0\) into the equation and solve for \(x\). This process helps us find the points where the graph of the equation intersects the x-axis.

Consider the equation given:
  • Substitute \(y = 0\): \(x^2(0) - x^2 + 4(0) = 0\).
  • Simplify the equation: \(-x^2 = 0\).
  • Solve for \(x\): We find \(x = 0\).
Therefore, the x-intercept is at the point (0, 0). At this point, the graph touches the x-axis. Finding x-intercepts is crucial in understanding the behavior of the graph along the x-axis, as it shows where the output (or y) is equal to zero.
y-intercepts
The y-intercepts of an equation indicate where the graph crosses the y-axis. On the y-axis, the value of x is zero. So to find y-intercepts, we set \(x = 0\) in the equation and solve for \(y\). This provides the points at which the graph intersects the y-axis.

For the given equation, the steps to find the y-intercepts are:
  • Substitute \(x = 0\): \(0^2y - 0^2 + 4y = 0\).
  • Simplify the equation: \(4y = 0\).
  • Solve for \(y\): We find \(y = 0\).
Thus, the y-intercept is also at the point (0, 0). This means that the graph touches the y-axis exactly at the origin. Identifying y-intercepts provides insight into where the graph begins vertically, especially important in sketching or analyzing graphs.
solving equations
Solving equations is a fundamental process in mathematics that involves finding the values of variables that make the equation true. For intercepts, we set one variable to zero in turn and solve for the other. This strategy simplifies the problem and results in finding where the graph crosses each axis.

In both cases for the given equation, the steps involved:
  • Substituting one variable to zero (either \(x\) or \(y\)).
  • Simplifying the resulting equation.
  • Solving for the remaining variable.
This procedure breaks down the problem into manageable parts, making it easier to solve. Understanding these steps is essential since finding intercepts is often one of the first tools used to sketch a graph, analyze functions, and gain insights into the behavior of complex equations. By mastering this process, students can tackle a wide range of problems in calculus and beyond.

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 a graphing utility to estimate the limit (if it exists). \(\lim _{x \rightarrow 2} \frac{x^{2}-5 x+6}{x^{2}-4 x+4}\)

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. \(\lim _{x \rightarrow 0^{-}} \frac{\frac{1}{x+4}-\frac{1}{4}}{x}\) \(\begin{array}{|c|c|c|c|c|c|}\hline x & {-0.5} & {-0.1} & {-0.01} & {-0.001} & {0} \\ \hline f(x) & {} & {} & {} & {?} \\ \hline\end{array}\)

Use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 0^{-}} \frac{x+1}{x}\)

Find the limit (if it exists). \(\lim _{t \rightarrow 1} \frac{t^{2}+t-2}{t^{2}-1}\)

use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one. If it is, find its inverse function. $$ g(t)=\frac{t+3}{1-t} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.