Problem 74 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=|x|-3\)

Step-by-Step Solution

Answer

The graph of the equation \(y = |x| - 3\) forms a 'V' shape, with the vertex at (0, -3) and is symmetrical about the Y-axis. The x-intercepts of the graph are at (-3, 0) and (3, 0) and the Y-intercept is at (0, -3).

Step 1: Graphing the equation

To sketch the graph of the equation \(y = |x| - 3\), understand that it is a linear equation with absolute value. At the point \(x = 0\), \(y\) will be equal to \(-3\). The graph will form a 'V' shape with the vertex on point (0, -3). Plot the point (0, -3). As \(x\) increases or decreases, \(y\) will increase. So plot a few more points like (1, -2), (-1, -2), (2, -1) etc. to sketch the curve.

Step 2: Identifying intercepts

The x-intercept occurs when \(y = 0\). So, set \(y = 0\) to get \(0 = |x| - 3\), which simplifies to \(x = 3\) and \(x = -3\). Therefore, there are two x-intercepts at points (-3, 0) and (3, 0). The Y-intercept occurs at \(x = 0\) which we know already, is at (0,-3).

Step 3: Test for symmetry

To test for symmetry, if the equation remains the same when replacing \(x\) with \(-x\), then it's symmetrical about the Y-axis. In the equation \(y = |x| - 3\), replacing \(x\) with \(-x\) yields \(y = |-x| - 3\), which simplifies back to \(y = |x| - 3\). Therefore, the graph is symmetrical about the Y-axis.

Key Concepts

X-Intercepts

An x-intercept is where the graph of an equation crosses the x-axis. This point shows what value of x makes the entire equation equal to zero. In this case, we have the function \( y = |x| - 3 \). To find the x-intercepts, we set \( y = 0 \) since the x-intercept occurs when the value of y is zero. So, we solve \( 0 = |x| - 3 \) which results in \( |x| = 3 \). This tells us that x can either be 3 or -3, giving us two points where the graph crosses the x-axis: \( (3, 0) \) and \( (-3, 0) \).

• Set \( y = 0 \) for x-intercepts.
• Solve \( |x| = 3 \) for values of x.

These intercepts are crucial for understanding where the graph intersects with the x-axis, indicating points of zero height for the function.

Y-Intercepts

Y-intercepts reveal where a graph crosses the y-axis. This is where x equals zero. For the equation \( y = |x| - 3 \), finding the y-intercept is straightforward. Substitute \( x = 0 \) into the equation. The calculation becomes:\[ y = |0| - 3 = -3\]Therefore, the y-intercept is the point (0, -3). This provides a starting point for the 'V' shape of the absolute value graph. Plotting this point is an essential step in sketching the graph. Remember:

• The y-intercept occurs when \( x = 0 \).
• Substitute \( x = 0 \) into the equation to find y.

Symmetry in Graphs

Symmetry in a graph helps simplify graphing and gives insight into the graph's characteristics. A graph is symmetrical about the y-axis if substituting \( -x \) for \( x \) results in the same equation. This is a common trait of functions involving absolute values.For \( y = |x| - 3 \), replace \( x \) with \( -x \) and see if the equation remains unchanged:\[y = |-x| - 3 = |x| - 3\]Since it remains the same, the graph is symmetrical about the y-axis. This means that if you have the left side of the graph, the right side will mirror it perfectly. Understanding symmetry can aid in checking work and saves time in graphing. It ensures accuracy and highlights the characteristic 'V' shape in absolute value functions.

• Check symmetry by replacing \( x \) with \( -x \).
• A symmetrical graph is mirrored along the y-axis.