Problem 30 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ x=|y| $$

Step-by-Step Solution

Answer

The x- and y-intercepts are both at the origin (0,0).

Step 1: Understanding the Equation

The given equation is \( x = |y| \). This means that for any value of \( y \), \( x \) will be the absolute value of \( y \). Hence, it is a relationship where \( x \) is always non-negative as it represents the absolute value of \( y \).

Step 2: Creating a Table of Values

To create a table of values, let's pick some values for \( y \), calculate \( x \), and then list them: - If \( y = -2 \), then \( x = |-2| = 2 \). - If \( y = -1 \), then \( x = |-1| = 1 \). - If \( y = 0 \), then \( x = |0| = 0 \). - If \( y = 1 \), then \( x = |1| = 1 \). - If \( y = 2 \), then \( x = |2| = 2 \). Thus, the table of values is: \[\begin{array}{c|c} y & x \ \hline -2 & 2 \ -1 & 1 \ 0 & 0 \ 1 & 1 \ 2 & 2 \end{array}\]

Step 3: Sketching the Graph

The equation \( x = |y| \) implies that for values of \( y \), \( x \) will mirror on both sides of the y-axis, from positive and negative to positive \( x \). Plot the points from the table, reflecting that for each positive \( y \), a corresponding \( x \) is the same as for negative \( y \). Thus, we plot points: (2, -2), (1, -1), (0,0), (1, 1), (2, 2). The graph is a symmetrical V-shape opening right, with the vertex at (0,0).

Step 4: Finding the X-Intercept

The \( x \)-intercept occurs where the graph crosses the \( x \)-axis. This happens when \( y = 0 \). Substituting \( y = 0 \) in the equation gives \( x = |0| = 0 \). So, the \( x \)-intercept is at (0,0).

Step 5: Finding the Y-Intercept

The \( y \)-intercept occurs where the graph crosses the \( y \)-axis, which happens when \( x = 0 \). Substituting \( x = 0 \) back into the equation \( x = |y| \), gives \( |y| = 0 \), meaning \( y = 0 \). So, the \( y \)-intercept is also at (0,0).

Key Concepts

Graphing Techniques

When graphing absolute value functions like the equation \( x = |y| \), it's important to understand the basic concepts of absolute value. An absolute value indicates the distance from zero on a number line, so it is always non-negative. This is why the graph for \( x = |y| \) is always on the right side of the y-axis. When you begin graphing, create a table of values that can help pinpoint key coordinates for the graph. Start by selecting both negative and positive values for \( y \), calculate their corresponding \( x \) values using the equation.

• The value of \( x \) will mirror for positive and negative \( y \) values.
• You'll plot symmetrical points to reflect this relationship.

Once you've plotted these points, you'll observe that the graph is a V-shape. It's symmetrical with respect to the x-axis because, for any \( y \) value and \( -y \) equivalent, the absolute values are equal. This shape opens to the right, with the tip or vertex at (0,0). The graph is unique due to its vertical symmetry, indicating equal values of \( x \) for both positive and negative \( y \) values.

X-Intercepts

Finding the x-intercept of the graph \( x = |y| \) involves identifying the point(s) where the graph crosses the x-axis. The x-axis represents the line \( y = 0 \), so to find the x-intercept, set \( y \) to zero in the equation. Substituting \( y = 0 \) into \( x = |y| \) gives us \( x = |0| = 0 \). Thus, the x-intercept of this graph is at the point (0,0).

This point (0,0) is crucial because it's the vertex of the V-shape. For the absolute value function \( x = |y| \), there is only one x-intercept. It indicates the smallest value of \( x \), as \( x \) reflects the non-negative distance from 0. In practical graphing scenarios, finding the x-intercept helps establish the anchor point from which you can extend the graph.

Y-Intercepts

The y-intercept of a graph is the point where the graph intersects the y-axis. For the equation \( x = |y| \), we find the y-intercept by setting \( x \) equal to 0. Plugging \( x = 0 \) into the equation \( x = |y| \), we solve for \( y \) by equating \( |y| = 0 \). Thus, \( y \) must also be 0, leading to the y-intercept at the point (0,0).

• This finding confirms the symmetry of the graph, as the vertex (0,0) serves as both the x-intercept and y-intercept.
• For this graph and others involving even absolute value equations, it is common to find intercepts at this singular vertex point.

The y-intercept is a key component, as it affirms the position of the vertex and is the minimum `x` value in y-direction symmetry. Hence, understanding intercepts clarifies how the graph behaves in relation to the axis, offering a foundational anchor for sketching and interpreting the absolute value function visually.