/*! 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 32 Graph the given equation. Label ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 32

Graph the given equation. Label each intercept. Use the concept of symmetry to confirm that the graph is correct. $$x=y^{2}-1$$

Short Answer

Expert verified
The intercepts of the graph of the function \(x = y^{2} - 1\) are (0,1), (0,-1) and (-1,0). It forms a parabola that opens to the left and right, and is symmetric about the y-axis.

Step by step solution

01

Plot the intercepts

The y-intercept is found by setting \(x = 0\), we get: \(0 = y^{2} - 1 \rightarrow y^{2} = 1\), giving \(y = 1\) and \(y = -1\). Hence, we have two y-intercepts, at points (0,1) and (0,-1). The x-intercept is found by setting \(y = 0\), we get: \(x = 0^{2} - 1 \rightarrow x = -1\). Hence, our x-intercept is (-1,0). So, plot these three points on the coordinate axes.
02

Complete the graph

Since the graph is symmetric, mirror the points on the right side of the y-axis over to the left side. Connect the points with a smooth curve, the graph of \(x = y^{2} - 1\) is a parabola that opens to the left and right, and it is symmetric about the y-axis.
03

Confirm the symmetry

To confirm the symmetry of the graph, check that each point plotted has a mirror image located directly opposite it on the other side of the y-axis. For a function to be symmetric with respect to the y-axis, a point \((a, b)\) on the graph should match with the point \((-a, b)\). In this case, for example if we take the point (0,1), its mirror image would be (0,-1) which is also on the graph. This confirms our 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 Intercepts
Intercepts are where a graph crosses the axes. They are crucial in sketching and analyzing graphs.
  • An **x-intercept** occurs where the graph touches or crosses the x-axis. Here, the y-value is zero.
  • A **y-intercept** is where the graph crosses the y-axis, making the x-value zero.

To find these intercepts in the equation, setup equations based on each intercept. Take the equation from the problem: \(x = y^2 - 1\). For the **y-intercepts**, set \(x = 0\) and solve for \(y\). This yields points \((0, 1)\) and \((0, -1)\).

For the **x-intercept**, set \(y = 0\) and solve for \(x\), giving the point \((-1, 0)\). These points give precise clues on where the graph cuts the axes.
Grasping Symmetry in Graphs
Symmetry simplifies understanding graphs by showing how parts of the graph relate.
  • A graph is **symmetric about the y-axis** if for every point \((x, y)\), there is a corresponding point \((-x, y)\).
  • Verify symmetry by checking if flipping points across the y-axis lands on another graph point.

In our exercise, with the graph of \(x = y^2 - 1\), we see symmetry about the y-axis. The points \((0,1)\) and \((0,-1)\) serve as a test. One point is simply the negative of the other on the y-axis.

Recognizing this type of symmetry helps in quickly sketching and verifying the accuracy of graphs.
Navigating the Coordinate System
Understanding the coordinate system is key in graphing functions. It's a grid of horizontal and vertical lines forming cells.
  • The **x-axis** runs left to right, while the **y-axis** goes up and down. The intersection at zero is the **origin** \((0, 0)\).
  • Every point on the graph is described in terms of \((x, y)\) coordinates.

When graphing the equation \(x = y^2 - 1\), first plot any intercepts. For our exercise, the intercepts provide a starting framework, which is \((-1, 0)\), \((0, 1)\), and \((0, -1)\). Each helps in elucidating the graph orientation and structure.

Graph this on the coordinate plane to see the pattern.
Identifying a Parabola
A parabola is a distinct U-shaped curve you often encounter in quadratic graphs. Often defined by equations involving \(y^2\) or \(x^2\).
  • The general form here is similar to \(x = y^2\) but shifted horizontally.
  • In our equation \(x = y^2 - 1\), the subtraction of 1 shifts the parabola left by one unit on the graph.

Parabolas manifest symmetry naturally, centered around a vertical or horizontal line. In our case, it鈥檚 symmetrical about the y-axis. When graphed, connect the plotted intercepts smoothly to form the parabola's curves.

Remember, how a parabola opens, aligns, and its orientation provide insights into solving and understanding quadratic-like functions.

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 notation \(\left.f(x)\right|_{a} ^{b}\) is used to denote the difference \(f(b)-f(a) .\) That is, $$\left.f(x)\right|_{a} ^{b}=f(b)-f(a)$$ Evaluate \(\left.f(x)\right|_{0} ^{b}\) for the given function \(f\) and the indicated values of \(a\) and \(b\). $$f(x)=2 x^{3}-3 x^{2}-x ;\left.f(x)\right|_{0} ^{2}$$

Solve by completing the square or by using the quadratic formula. $$\sqrt{2} x^{2}+3 x+\sqrt{2}-0$$

Solve by completing the square or by using the quadratic formula. $$-x^{2}-7 x-1$$

The perimeter of a rectangle is 34 feet and its area is 60 square feet. Find the length and the width of the rectangle.

Given the points \(P_{1}(-3,4)\) and \(P_{2}(2,-4),\) evaluate \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.