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Magazine Chapter 2: Problem 29
\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=|x| $$
Short Answer
Expert verified
Both the x- and y-intercepts are at (0, 0).
Step by step solution
01
Choose values for x
To create a table of values and sketch the graph of the equation \( y = |x| \), we start by choosing several values for \( x \), both positive, negative, and zero to plot the respective \( y \) values. Commonly used \( x \) values are -3, -2, -1, 0, 1, 2, and 3.
02
Calculate corresponding y values
For each chosen \( x \) value, calculate the corresponding \( y \) by applying \( y = |x| \). This results in \( y = |-3| = 3 \), \( y = |-2| = 2 \), \( y = |-1| = 1 \), \( y = |0| = 0 \), \( y = |1| = 1 \), \( y = |2| = 2 \), and \( y = |3| = 3 \). Tabulate these \( (x, y) \) pairs.
03
Tabulate (x, y) values
Using the results from Step 2, create a table:| \(x\) | \(y\) ||----|----||-3 | 3 ||-2 | 2 ||-1 | 1 || 0 | 0 || 1 | 1 || 2 | 2 || 3 | 3 |
04
Graph the function
Plot the points from the table in a coordinate system. Connect these points with straight lines. The graph should form a V-shape with its lowest point at the origin (0, 0).
05
Find x- and y-intercepts
The x-intercept and y-intercept are points where the graph crosses the x-axis and y-axis, respectively. For \( y = |x| \), substituting \( y = 0 \) and \( x = 0 \) both lead to the point (0, 0), so both intercepts are at (0, 0).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Table of Values
Creating a table of values is a fundamental step when it comes to graphing functions, like the absolute value function. By selecting a set of values for the variable \( x \), you can determine the corresponding \( y \) values. For the function \( y = |x| \), it's important to include a mix of negative, positive, and zero values for \( x \). This ensures we capture the behavior of the absolute value function on both sides of the y-axis.
For example:
For example:
- Select \( x \) values such as -3, -2, -1, 0, 1, 2, and 3.
- Apply the formula \( y = |x| \) to find \( y \) for each \( x \).
- Record pairs: (-3, 3), (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2), (3, 3).
X-Intercept
The x-intercept is the point on the graph where the line crosses the x-axis. At this point, the value of \( y \) is zero. For the function \( y = |x| \), we set \( y = 0 \) to find the x-intercept.
When \( y = 0 \), the equation \( y = |x| \) simplifies to \( 0 = |x| \). This means \( x \) must be 0, as the absolute value of zero is zero. Hence, the x-intercept for graphing \( y = |x| \) is at the point (0, 0). This fundamental intercept gives us a clear point of reference for sketching the V-shape of the graph.
When \( y = 0 \), the equation \( y = |x| \) simplifies to \( 0 = |x| \). This means \( x \) must be 0, as the absolute value of zero is zero. Hence, the x-intercept for graphing \( y = |x| \) is at the point (0, 0). This fundamental intercept gives us a clear point of reference for sketching the V-shape of the graph.
Y-Intercept
Similarly, the y-intercept is the point where the graph crosses the y-axis, which is the case when \( x = 0 \). For any function, you find the y-intercept by substituting \( x = 0 \) into the equation.
For \( y = |x| \), substituting \( x = 0 \) gives \( y = |0| = 0 \). Thus, the y-intercept is also at the point (0, 0). This is an example of a vertex function, where both intercepts coincide, highlighting the symmetry of absolute value graphs.
This intercept is a key feature that helps to clearly define and sketch the overall shape of the graph, marking the minimum point of the V-shaped graph.
For \( y = |x| \), substituting \( x = 0 \) gives \( y = |0| = 0 \). Thus, the y-intercept is also at the point (0, 0). This is an example of a vertex function, where both intercepts coincide, highlighting the symmetry of absolute value graphs.
This intercept is a key feature that helps to clearly define and sketch the overall shape of the graph, marking the minimum point of the V-shaped graph.
Sketching Graphs
When sketching graphs of absolute value functions like \( y = |x| \), using the table of values and identified intercepts is very helpful. Begin by plotting the points from the table onto a coordinate grid. These points will include the x- and y-intercepts which we found to be (0, 0).
Once the points are plotted:
Once the points are plotted:
- Draw straight lines connecting the points on either side of the origin.
- The resulting shape should look like a "V" centered at the origin.
- The arms of the V will open symmetrically, one going upwards to the right and the other to the left.
