Problem 14 - Precalculus

Question

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=|x|-2$$

Step-by-Step Solution

Answer

The x-intercepts are \((2, 0)\) and \((-2, 0)\), the y-intercept is \((0, -2)\), and the graph is symmetric about the y-axis.

Step 1: Identify the x-intercept

To find the x-intercept, set the equation equal to zero: \(y = |x| - 2 = 0\). Solve for \(x\): \(|x| = 2\). This gives us two solutions, \(x = 2\) and \(x = -2\). Thus, the x-intercepts are at \((2, 0)\) and \((-2, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\): \(y = |0| - 2 = -2\). Therefore, the y-intercept is at \((0, -2)\).

Step 3: Test for symmetry with respect to the y-axis

For y-axis symmetry, check if \(f(-x) = f(x)\). Evaluate \(f(-x) = |-x| - 2 = |x| - 2 = f(x)\). Since this condition is satisfied, the graph is symmetric with respect to the y-axis.

Step 4: Graph the equation

Plot the intercepts \((2, 0)\), \((-2, 0)\), and \((0, -2)\) on the coordinate plane. Draw the V-shape that opens upwards, with its vertex at \((0, -2)\) due to the absolute value function. The graph will reflect symmetry along the y-axis, which can be seen as both sides of the V are mirror images across the y-axis.

Key Concepts

Understanding x-intercepts

The x-intercepts of a graph are the points where the graph touches or crosses the x-axis. To find them, you set the output of the function, or the y-value, equal to zero. For the equation given, this step involves solving:
\[ y = |x| - 2 = 0 \]
This equation implies that \[|x| = 2\].
Absolute value equations can have two solutions because they account for both the positive and negative scenarios. Therefore, we get two different values for x:

• \( x = 2 \)
• \( x = -2 \)

These solutions indicate the x-intercepts are \((2, 0)\) and \((-2, 0)\).
Their significance is that at these points, the function crosses the x-axis, making the function value zero. Always remember, solving for x-intercepts often involves considering the absolute values, ensuring no potential solutions are missed.

Finding y-intercepts

Y-intercepts represent the points where the graph crosses the y-axis. To find the y-intercept of a function, set the input value, or x, to zero, because this point is where the line connects with the y-axis. Using the equation provided, we substitute zero for x:
\[ y = |0| - 2 = -2 \]
Thus, the y-intercept is at \((0, -2)\).
This tells us, when x equals zero, the function value drops to -2.The position of the y-intercept is crucial as it gives us critical information about the vertical shift of the graph. Specifically, for our function, it indicates the lowest point, or vertex of our V-shaped graph, lies two units below the x-axis. Y-intercepts help in visualizing the initial shape and shift of the graph particularly in absolute value functions.

Exploring y-axis symmetry

Y-axis symmetry in a graph means that one side is a mirror image of the other across the y-axis. To test for this symmetry in a function, substitute \(-x\) for \(x\) in the equation and check if the equation remains unchanged. For our function:
\[ f(-x) = |-x| - 2 = |x| - 2 = f(x) \]
This confirms that the equation is unchanged, thus verifying that the graph is symmetric about the y-axis.This symmetry is particularly useful while graphing, as you do not need to calculate points separately on either side of the y-axis. A V-shaped graph, like this one, clearly reflects symmetry, as both arms of the 'V' are equally spread on either side of the axis. This means any point on the left of the y-axis has a corresponding mirrored point on the right. Symmetry makes it easier to predict the shape and behaviour of the graph, and it provides visual balance.