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Magazine Chapter 2: Problem 19
19鈥�44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=-x $$
Short Answer
Expert verified
The x- and y-intercepts are both at (0, 0), and the graph of \( y = -x \) is symmetric about the origin.
Step by step solution
01
Create a Table of Values
Choose a set of x-values and calculate the corresponding y-values using the equation \( y = -x \). A simple set could be \( x = -2, -1, 0, 1, 2 \). Calculate:- For \( x = -2 \), \( y = -(-2) = 2 \).- For \( x = -1 \), \( y = -(-1) = 1 \).- For \( x = 0 \), \( y = -(0) = 0 \).- For \( x = 1 \), \( y = -(1) = -1 \).- For \( x = 2 \), \( y = -(2) = -2 \).Use these points to make a table:\[\begin{array}{c|c}x & y \\hline-2 & 2 \-1 & 1 \0 & 0 \1 & -1 \2 & -2 \\end{array}\]
02
Sketch the Graph
Plot the points from the table onto a coordinate plane:- Plot (鈭�2, 2)- Plot (鈭�1, 1)- Plot (0, 0)- Plot (1, 鈭�1)- Plot (2, 鈭�2)Draw a line through these points, which represents the graph of \( y = -x \). The line should extend through the origin at a 45-degree angle to both axes.
03
Find the Intercepts
To find the y-intercept, set \( x = 0 \) in the equation \( y = -x \), which gives \( y = 0 \). Thus, the y-intercept is at \( (0, 0) \).For the x-intercept, set \( y = 0 \) in the equation \( y = -x \), which gives \( x = 0 \). Therefore, the x-intercept is also at \( (0, 0) \).
04
Test for Symmetry
Check symmetry with respect to the y-axis, x-axis, and origin:- Y-axis: Substitute \( x \) with \( -x \). The equation becomes \( y = x \), not equivalent to the original.- X-axis: Substitute \( y \) with \( -y \). The equation becomes \( -y = -x \), which is equivalent to the original \( y = -x \), indicating no x-axis symmetry.- Origin: Substitute both \( x \) with \( -x \) and \( y \) with \( -y \). The equation becomes \( -y = x \), which simplifies to the original \( y = -x \), confirming origin symmetry. Thus, the graph is symmetric about the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding x-intercepts
The x-intercept of a graph is the point where the curve crosses the x-axis. To find the x-intercept of a linear equation like \( y = -x \), we focus on setting \( y = 0 \) and solving for \( x \). This is because the x-axis corresponds to \( y = 0 \).
- For our equation \( y = -x \), set \( y = 0 \): \( 0 = -x \).
- This rearranges to \( x = 0 \).
Exploring y-intercepts
Y-intercepts are crucial for recognizing where a graph crosses the y-axis. For the linear equation \( y = -x \), the y-axis intercept occurs when \( x = 0 \).
- Inserting \( x = 0 \) into the equation yields: \( y = -(0) = 0 \).
Understanding symmetry in graphs
Symmetry can tell us a lot about the balance and pattern in a graph. In the context of \( y = -x \), we analyze different types of symmetry by testing transformations.
Types of Symmetry
- Y-axis symmetry: For symmetry around the y-axis, replacing \( x \) with \( -x \) should yield the same equation, but \( y = x \) results, showing no symmetry.
- X-axis symmetry: Testing x-axis symmetry with \( y = -y \) and replacing gives \(-y = -x\); this isn't symmetric across the x-axis.
- Origin symmetry: Checking for origin symmetry involves replacing both \( x \) with \( -x \) and \( y \) with \( -y \). This results in \(-y = x\), simplifying back to \( y = -x \), confirming symmetry about the origin.
Creating a table of values
A table of values is a convenient method for plotting a linear equation. By substituting different values of \( x \) into the equation \( y = -x \), we calculate corresponding \( y \)-values.
- For example, choose \( x \, values\) like \( -2, -1, 0, 1, 2 \).
- Calculate: \[ \begin{align*} \text{When } x = -2, & \ y = -(-2) = 2 \ \text{When } x = -1, & \ y = -(-1) = 1 \ \text{When } x = 0, & \ y = -(0) = 0 \ \text{When } x = 1, & \ y = -(1) = -1 \ \text{When } x = 2, & \ y = -(2) = -2 \end{align*} \]
