/*! 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 66 Find any intercepts of the graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 66

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. \(x+3=|y-5|\)

Short Answer

Expert verified
The x-intercept is (2, 0) and y-intercepts are (0, 8) and (0, 2). No symmetry with respect to x-axis, y-axis, or origin.

Step by step solution

01

Identify x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\): \[x + 3 = |0 - 5|\] \[x + 3 = 5\] \[x = 2\]. Hence, the x-intercept is \((2, 0)\).
02

Identify y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation and solve for \(y\): \[0 + 3 = |y - 5|\] \[3 = |y - 5|\] This gives two possible equations: \(y - 5 = 3\) and \(y - 5 = -3\). Solving them, we get \(y = 8\) and \(y = 2\). Thus, the y-intercepts are \((0, 8)\) and \((0, 2)\).
03

Test for x-axis symmetry

To check for symmetry with respect to the x-axis, replace \(y\) with \(-y\) and see if it results in the same equation: \[x + 3 = |-y - 5|\]. This is not the same as the original equation \(x+3=|y-5|\), so there's no x-axis symmetry.
04

Test for y-axis symmetry

To check for symmetry with respect to the y-axis, replace \(x\) with \(-x\) and see if it results in the same equation: \[-x + 3 = |y - 5|\]. This is not the same as the original equation \(x+3=|y-5|\), so there's no y-axis symmetry.
05

Test for symmetry about the origin

To check for symmetry with respect to the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) and see if it results in the same equation: \[-x + 3 = |-y - 5|\]. This is neither equivalent to the original equation. Thus, there's no origin symmetry.

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 the x-intercept
The x-intercept of a graph is the point where the graph crosses the x-axis. This is the point at which the value of y is zero. To find the x-intercept for a given equation, simply set y to 0 and solve for x.
In our equation, \(x + 3 = |y - 5|\), we substitute y with 0. This gives us \(x + 3 = |0 - 5|\), which simplifies to \(x + 3 = 5\). Solving for x, we find \(x = 2\). Thus, the x-intercept of this equation is the point \((2, 0)\).
Whenever you have an equation and need to find the x-intercept, remember:
  • Set \(y = 0\)
  • Solve for \(x\)
  • The solution gives you the x-intercept as \((x, 0)\)
Exploring the y-intercept
The y-intercept is where the graph touches the y-axis, meaning it's the point where x is zero. To find the y-intercept, you set x to 0 in the equation and solve for y.
Let's look back at our equation, \(x + 3 = |y - 5|\). By setting x to 0, we have \(0 + 3 = |y - 5|\) or simply \(3 = |y - 5|\). This absolute equation gives two potential solutions: \(y - 5 = 3\) and \(y - 5 = -3\). Solving these, we find \(y = 8\) and \(y = 2\). Hence, the y-intercepts are \((0, 8)\) and \((0, 2)\).
When you're tasked with finding the y-intercept:
  • Set \(x = 0\)
  • Solve the resulting equation for \(y\)
  • Record the y-intercepts as \((0, y)\)
Graph symmetry overview
Graph symmetry helps us understand the overall shape and balance of the graph. A graph can have different types of symmetry such as x-axis symmetry, y-axis symmetry, or symmetry about the origin.
Checking for symmetry involves substituting values in the equation and checking if it remains unchanged:
  • x-axis symmetry: Replace \(y\) with \(-y\).
  • y-axis symmetry: Replace \(x\) with \(-x\).
  • Origin symmetry: Replace \(x\) with \(-x\) and \(y\) with \(-y\).
Observing the behaviors of graphs relative to these practices help indicate the presence or absence of symmetry.
Understanding x-axis symmetry
A graph exhibits x-axis symmetry if the graph mirrors itself along the x-axis. This means if you replace \(y\) with \(-y\), the equation remains the same.
Applying this to our equation \(x + 3 = |y - 5|\), we check for x-axis symmetry by examining \(x + 3 = |-y - 5|\). Since this is not the same as the original equation, it indicates that there is no x-axis symmetry.
This quick substitution can easily show whether or not x-axis symmetry exists:
  • Change \(y\) to \(-y\)
  • If the equation doesn't change, x-axis symmetry is present. Otherwise, it's absent.
Y-axis symmetry explained
Y-axis symmetry occurs when a graph is a mirror image over the y-axis. You find y-axis symmetry by replacing \(x\) with \(-x\) in the equation and checking if the equation remains unchanged.
In our equation \(x + 3 = |y - 5|\), replacing \(x\) with \(-x\) results in \(-x + 3 = |y - 5|\). Since this is not identical to the original equation, the graph lacks y-axis symmetry.
To check for y-axis symmetry effectively:
  • Substitute \(x\) with \(-x\)
  • See if the equation is consistent; if not, there's no y-axis symmetry.

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

The equation \((x-4)^{2}+(y+10)^{2}=0\) does not describe a circle. What is the graph of this equation?

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{(2 x+1)^{2}-9}{x-1}\) (b) \(\lim _{x \rightarrow 1} \frac{(2 x+1)^{2}-9}{x-1}\)

Use rationalization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{25-t}{5-\sqrt{t}}\) (b) \(\lim _{t \rightarrow 25} \frac{25-t}{5-\sqrt{t}}\)

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). $$ \text { (a) } \frac{x^{2}-8 x}{x^{2}-6 x-16} $$ $$ \text { (b) } \lim _{x \rightarrow 8} \frac{x^{2}-8 x}{x^{2}-6 x-16} $$

In Problems \(31-34\), sketch the set of points in the \(x y\) plane whose coordinates satisfy the given inequality. \(x^{2}+y^{2} \geq 9\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.