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Magazine Chapter 1: Problem 59
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
Rearrange the given equation to express y in terms of x. Starting with the equation: \[ 2x - y = 6 \]Add \(y\) to both sides: \[ 2x = y + 6 \]Now, subtract 6 from both sides to solve for \(y\): \[ y = 2x - 6 \]
02
Make a table of values
Create a table of values by choosing different values for \(x\) and calculating the corresponding \(y\) values using the equation \(y = 2x - 6\).| \(x\) | \(y\) ||----|----|| 0 | -6 || 1 | -4 || 2 | -2 || 3 | 0 || 4 | 2 |These points \((0, -6), (1, -4), (2, -2), (3, 0), (4, 2)\) will help in plotting the graph.
03
Plot the graph
Using the table of values, plot the points on a graph and connect them with a straight line. This line represents the equation \(y = 2x - 6\).
04
Find x-intercept
The \(x\)-intercept occurs where \(y = 0\). Substitute \(y = 0\) into the equation \(2x - y = 6\) and solve for \(x\):\[ 2x - 0 = 6 \]\[ 2x = 6 \]\[ x = 3 \]The \(x\)-intercept is \((3, 0)\).
05
Find y-intercept
The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation \(y = 2x - 6\) to find \(y\):\[ y = 2(0) - 6 \]\[ y = -6 \]The \(y\)-intercept is \((0, -6)\).
06
Test for symmetry
To test for symmetry about the y-axis, check if replacing \(x\) with \(-x\) gives the same equation. Substitute \(-x\) for \(x\) in \(y = 2x - 6\):\[ y = 2(-x) - 6 = -2x - 6 \]Since this is not the same equation, the graph is not symmetric about the y-axis.To test for symmetry about the x-axis, replace \(y\) with \(-y\):\[ -y = 2x - 6 \]This results in a different equation, so there is no symmetry about the x-axis.Finally, for origin symmetry, check \(y = 2(-x) - 6\) by comparing with \(-y = 2x - 6\). These are not equivalent, so there is no origin symmetry.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
x-intercept
The x-intercept of a graph is the point where the graph crosses the x-axis. At this point, the value of \( y \) is always zero. To find the x-intercept of a linear equation, you substitute zero for \( y \) in the given equation and solve for \( x \).
For the equation \( 2x - y = 6 \), you set \( y = 0 \) and solve:
- \( 2x - 0 = 6 \)
- Simplifying that gives \( 2x = 6 \)
- Dividing by 2 results in \( x = 3 \)
Therefore, the x-intercept is the point \((3, 0)\). This means the line goes through the x-axis at 3.
Understanding the x-intercept is essential for sketching graphs as it provides a point where the line meets the x-axis, helping to verify accuracy when drawing.
For the equation \( 2x - y = 6 \), you set \( y = 0 \) and solve:
- \( 2x - 0 = 6 \)
- Simplifying that gives \( 2x = 6 \)
- Dividing by 2 results in \( x = 3 \)
Therefore, the x-intercept is the point \((3, 0)\). This means the line goes through the x-axis at 3.
Understanding the x-intercept is essential for sketching graphs as it provides a point where the line meets the x-axis, helping to verify accuracy when drawing.
y-intercept
The y-intercept of a graph is where it crosses the y-axis. At this point, the value of \( x \) is zero. To find the y-intercept of a linear equation, substitute zero for \( x \) and solve for \( y \).
Let's find the y-intercept for \( y = 2x - 6 \):
- Substitute \( x = 0 \) into the equation
- \( y = 2(0) - 6 \)
- Thus, \( y = -6 \)
This calculation shows that the y-intercept is \((0, -6)\). So the graph intersects the y-axis at -6.
Identifying the y-intercept is crucial for creating an accurate graph, as it offers a reference point that helps define the line's position relative to the axes.
Let's find the y-intercept for \( y = 2x - 6 \):
- Substitute \( x = 0 \) into the equation
- \( y = 2(0) - 6 \)
- Thus, \( y = -6 \)
This calculation shows that the y-intercept is \((0, -6)\). So the graph intersects the y-axis at -6.
Identifying the y-intercept is crucial for creating an accurate graph, as it offers a reference point that helps define the line's position relative to the axes.
graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and then drawing a straight line through these points. This line visually represents all the solutions of the equation.
First, rearrange the linear equation into slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the equation \( 2x - y = 6 \), rearranging gives us \( y = 2x - 6 \).
With this equation:
Using these points, like \((0, -6)\), \((1, -4)\), and \((3, 0)\), you can sketch the line. A linear equation always forms a straight line, making graphing a way to visualize the function clearly.
First, rearrange the linear equation into slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the equation \( 2x - y = 6 \), rearranging gives us \( y = 2x - 6 \).
With this equation:
- The slope \( m \) is 2, indicating how steep the line is. For each unit increase in \( x \), \( y \) increases by 2.
- The y-intercept \( b \) is -6, showing the line crosses the y-axis at -6.
Using these points, like \((0, -6)\), \((1, -4)\), and \((3, 0)\), you can sketch the line. A linear equation always forms a straight line, making graphing a way to visualize the function clearly.
symmetry in graphs
Symmetry in graphs shows how a graph replicates its shape in one part when reflected over a line or point. Common types of symmetry include y-axis symmetry, x-axis symmetry, and origin symmetry.
For y-axis symmetry, substituting \( -x \) for \( x \) should yield the original equation. With \( y = 2x - 6 \), replacing \( x \) gives \( y = -2x - 6 \), which isn't identical to the original. Thus, there's no y-axis symmetry.
Testing for x-axis symmetry involves replacing \( y \) with \( -y \) and checking if the equation remains unchanged. Here, \( -y = 2x - 6 \) modifies the equation, indicating no x-axis symmetry.
For origin symmetry, both \( x \) and \( y \) are replaced with their negatives. Since substituting produces different equations, \( y = 2(-x) - 6 \) and \( -y = 2x - 6 \), there's no origin symmetry.
Understanding symmetry helps identify predictable patterns in graphs, easing the process of sketching and interpreting these graphical solutions.
For y-axis symmetry, substituting \( -x \) for \( x \) should yield the original equation. With \( y = 2x - 6 \), replacing \( x \) gives \( y = -2x - 6 \), which isn't identical to the original. Thus, there's no y-axis symmetry.
Testing for x-axis symmetry involves replacing \( y \) with \( -y \) and checking if the equation remains unchanged. Here, \( -y = 2x - 6 \) modifies the equation, indicating no x-axis symmetry.
For origin symmetry, both \( x \) and \( y \) are replaced with their negatives. Since substituting produces different equations, \( y = 2(-x) - 6 \) and \( -y = 2x - 6 \), there's no origin symmetry.
Understanding symmetry helps identify predictable patterns in graphs, easing the process of sketching and interpreting these graphical solutions.
