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Chapter 2: Problem 23

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. $$ 2 x-y=6 $$

Short Answer

Expert verified
The x-intercept is (3,0), the y-intercept is (0,-6), and there is no symmetry.

Step by step solution

01

Solve for y

First, start by solving the equation \(2x - y = 6\) for \(y\). Rearrange the equation to isolate \(y\) on one side.\[ y = 2x - 6 \].
02

Create a Table of Values

Choose several values of \(x\) to find the corresponding \(y\) values using the equation \(y = 2x - 6\). For example: - If \(x = 0\), \(y = 2 \times 0 - 6 = -6\). - If \(x = 3\), \(y = 2 \times 3 - 6 = 0\). - If \(x = 5\), \(y = 2 \times 5 - 6 = 4\).This results in the points \((0, -6)\), \((3, 0)\), and \((5, 4)\).
03

Determine Intercepts

To find the y-intercept, set \(x = 0\) and solve for \(y\):\(y = 2(0) - 6 = -6\). Thus, the y-intercept is \((0, -6)\).To find the x-intercept, set \(y = 0\) and solve for \(x\):\[ 0 = 2x - 6\] \[ 2x = 6 \] \[ x = 3\]. Thus, the x-intercept is \((3, 0)\).
04

Sketch the Graph

Using the points from the table and the intercepts, draw the line on a graph. The line passes through \((0, -6)\), \((3, 0)\), and \((5, 4)\). This line will have a positive slope.
05

Test for Symmetry

The line equation \(y = 2x - 6\) is not symmetric about the y-axis, x-axis, or the origin since substituting \(-x\) for \(x\) or \(-y\) for \(y\) does not yield the original equation. Therefore, there is no symmetry.

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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 line crosses the x-axis. At this point, the y-value is zero. To find the x-intercept for the equation \[2x - y = 6,\] we set y to zero and solve for x.
  • Start with the equation: \( 0 = 2x - 6 \).
  • Rearrange it: \( 2x = 6 \).
  • Divide both sides by 2: \( x = 3 \).
The x-intercept is the point where the graph touches the x-axis at \((3,0)\). Understanding x-intercepts can help you quickly identify important graph points, especially when sketching lines.
Discovering the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. Here, the x-value is zero. To determine the y-intercept for our equation \[2x - y = 6,\] set x to zero, and solve for y.
  • Plug 0 in for x: \( y = 2(0) - 6 \).
  • Calculate y: \( y = -6 \).
Thus, the y-intercept is at \((0, -6)\). This point helps in understanding how the graph interacts with the y-axis. It's a key reference point when drawing the line.
Exploring Symmetry in Graphs
Symmetry in graphing refers to how one side of the graph mirrors another. The line equation \(y = 2x - 6\) is checked for symmetry about different axes or the origin by substituting values:
  • Check for y-axis symmetry by replacing \(x\) with \(-x\) and see if the equation remains unchanged.
  • Try x-axis symmetry by replacing \(y\) with \(-y\).
  • Check for origin symmetry by replacing \(x\) with \(-x\) and \(y\) with \(-y\).
In this case, none of these substitutions result in the same original equation, \(2x - y = 6\). Therefore, the graph is not symmetric about any of these axes. Recognizing symmetry can simplify graphing and analysis but is not always present in linear equations.
Creating a Table of Values
When graphing linear equations like \(y = 2x - 6\), a table of values is a helpful tool. By calculating several coordinated pairs through the equation, you get anchor points for drawing the line.
  • Choose values for x, for instance, \(x = 0, 3, \text{and } 5\).
  • Calculate corresponding y values: \(y = -6, 0, \text{and } 4\) respectively.
This results in pairs: \((0, -6), (3, 0), \text{and } (5, 4)\). Plotting these points on a graph and connecting them will give a visual representation of the equation. This method supports accuracy and understanding of how linear equations appear graphically.

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