Problem 24 - College Algebra

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1\) \(0,1,2,\) and 3. $$y=|x|-1$$

Step-by-Step Solution

Answer

After calculating and plotting the points for each x-value on the plane, and connecting these points, get a V-shaped curve shifted downward by 1 unit, which is the graph of the function \(y=|x|-1\).

Step 1: Understand the absolute function

Understanding the key features of an absolute function is crucial. The absolute function \(y=|x|\) is a V-shaped function, which touches the x-axis at the origin (0,0) and points upwards. It is symmetrical about the y-axis.

Step 2: Compute the y-values

Plug each x-value into the equation \(y=|x|-1\) to find the corresponding y-values. For example, when \(x=-3\), \(y=|-3|-1=2\). Do it for all x-values: -3, -2, -1, 0, 1, 2, and 3.

Step 3: Plot the points

Now, with the calculated coordinates, you can plot them on a graph. These points should be on the V-shaped curve of the absolute function, which is shifted down by 1.

Step 4: Draw the function graph

Connect all the plotted points with a smooth curve to represent the function \(y=|x|-1\). The graph should resemble the graph of \(y=|x|\) function but shifted downward by 1 unit.

Key Concepts

Graphing Equations

When faced with graphing equations, the goal is to visually represent the relationship between variables, usually on a coordinate plane. In this exercise, the equation given is \(y = |x| - 1\).
To graph such an equation:

• Understand the type of equation or function. Here, you are dealing with an absolute value function, which is known for its V-shaped graph.
• Determine any transformations that occur. For instance, subtracting 1 from \(|x|\) shifts the graph of \(y = |x|\) one unit downward.

Finding these transformations helps in accurately positioning the graph on the plane. As you plot points in the next steps, keep these transformations in mind for precise graph placement.

Plotting Points

Plotting points is a crucial step in graphing equations. It involves calculating coordinates and marking them on the graph.
For the equation \(y = |x| - 1\), follow these steps to plot points:

• Choose a range of x-values, as suggested in the exercise: -3, -2, -1, 0, 1, 2, and 3.
• Substitute these x-values into the equation to compute their corresponding y-values.

Example calculations include:

• For \(x = -3\), compute \(y = |-3| - 1 = 2\), resulting in the point (-3, 2).
• Continue this process for each x-value: (-2, 1), (-1, 0), etc.

These calculated (x, y) pairs are then plotted as points on the graph. Connecting these points will begin to form the characteristic V shape of the absolute value function.

V-shaped Graph

The V-shaped graph is a hallmark of absolute value functions like \(y = |x|\). In this exercise, the graph we are focusing on is \(y = |x| - 1\).
The initial graph of \(y = |x|\) has a symmetrical V shape, with the point (0,0) as its vertex, located on the x-axis. It's symmetrical due to the nature of absolute value, which measures the non-negative distance of a number from zero.

• For the function \(y = |x| - 1\), this V shape is moved downward by 1 unit. The vertex of the graph is now at the point (0, -1).
• This transformation doesn't change the symmetry—still balanced around the y-axis—nor the angle of the V.

Understanding the structure of the V-shaped graph allows you to predict the graph's behavior and assists in sketching it accurately by connecting the points you plotted. This gives a complete visualization of the function's rule, helping emphasize the graph's unique characteristics.