Problem 38 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ x=|y| $$

Step-by-Step Solution

Answer

The x- and y-intercepts are both at the origin (0,0). The graph is symmetric about the x-axis.

Step 1: Understand the Equation

The equation given is in the form \(x = |y|\). This represents a relation where the value of \(x\) is the absolute value of \(y\). For each \(y\), \(x\) will have a non-negative value.

Step 2: Create a Table of Values

Select a range of \(y\) values, calculate \(x = |y|\) for each value, and create a table. Here is the table:| \(y\) | \(x = |y|\) ||-------|-------------||-3 | 3 ||-2 | 2 ||-1 | 1 || 0 | 0 || 1 | 1 || 2 | 2 || 3 | 3 |The table shows that for negative \(y\), the \(x\) value is the same as for the positive \(y\).

Step 3: Sketch the Graph

Plot the points from the table on a coordinate plane. The graph forms a V-shape with two lines intersecting at the origin (0,0). The left line goes downward and to the right, while the right line goes upward and to the right.

Step 4: Find the x- and y-intercepts

To find the y-intercept, set \(x = 0\) and solve for \(y\), which gives \(y = 0\). Thus, the y-intercept is at (0,0). For the x-intercept, set \(y = 0\) and solve for \(x\), which also gives \(x = 0\). Thus, the x-intercept is at (0,0). Both intercepts are at the origin.

Step 5: Test for Symmetry

Check for symmetry:- **Symmetry about the x-axis**: Substitute \(-y\) into the equation. It results in \(x = |-y| = |y|\), confirming symmetry around the x-axis.- **Symmetry about the y-axis**: Replace \(-x\) into the equation, which results in an invalid equation (\(-x = |y|\)), so no symmetry around the y-axis.- **Symmetry about the origin**: Substituting \(-x\) and \(-y\) simultaneously leads to \(-x = |-y|\), which is not valid (indicates no symmetry about the origin). Thus, it is only symmetric about the x-axis.

Key Concepts

x-intercepts

In the context of the graph of an equation, an x-intercept is a point where the graph crosses or touches the x-axis. At this point, the value of the y-coordinate is zero.
For the equation \(x = |y|\), the x-intercept can be found by setting \(y = 0\) and solving for \(x\). Consequently, when \(y = 0\), \(x\) becomes \(|0|\) which is 0. Therefore, the x-intercept for this particular graph is at the origin, which is the point (0, 0).
As x-intercepts are vital in understanding where functions interact with the x-axis, recognizing the role of the origin point can provide insights into the characteristics of the graph.

y-intercepts

y-intercepts serve as the points where the graph intersects the y-axis, and at these points, the x-coordinate is always zero.
In terms of the equation \(x = |y|\), we determine the y-intercept by setting \(x = 0\) and solving for \(y\). This equation implies that \(0 = |y|\), meaning \(y\) must also be zero. Hence, the y-intercept, similar to the x-intercept, occurs at the origin, noted as the point (0, 0).
Identifying y-intercepts is essential as it helps when graphing functions and recognizing how the equation behaves in relation to the axes. In our case, the graph touches the y-axis only at one place: the origin.

graph symmetry

Symmetry in graphing illustrates how a graph can mirror itself across axes or through points.
For the equation \(x = |y|\), let's examine the graph symmetry:

• Symmetry about the x-axis: Substitute \(-y\) for \(y\) in the equation to see if the equation remains unchanged. Replacing \(y\) with \(-y\) results in \(x = |-y| = |y|\), confirming symmetry across the x-axis.
• Symmetry about the y-axis: Replace \(x\) with \(-x\) in the equation, resulting in \(-x = |y|\). This does not hold true because \(-x\) does not equal \(x\), so there's no symmetry along the y-axis.
• Symmetry about the origin: When simultaneously substituting \(-x\) for \(x\) and \(-y\) for \(y\), the equation becomes \(-x = |-y|\). Like before, this transformation is invalid, indicating a lack of symmetry about the origin.

In summary, the graph of \(x = |y|\) is symmetric solely about the x-axis, creating a V-shaped graph where each side mirrors the other in relation to the horizontal axis.