Problem 24 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The graph will be a 'V' shape positioned below the x-axis by a distance of 1. The exact points on the graph will be (-3,2), (-2,1), (-1,0), (0,-1), (1,0), (2,1), and (3,2).

Step 1: Compute y for each x

Use the equation \(y = |x| - 1\) for each given value of x. This will yield corresponding y-values for each x-value.

Step 2: Plot each (x,y) pair.

On an x-y graph, mark a point for each (x,y) pair. This will result in a series of dots or points.

Step 3: Draw the Graph

After plotting all the points, connect them to form the graph. The shape should resemble a 'V', characteristic of absolute value functions, with the tip being at the point where \(x = 0\).

Key Concepts

Plotting Points on a Graph

Plotting points on a graph is like creating a map using coordinates. Each point on a graph represents an ordered pair (x, y). Here’s how to do it step-by-step.

• Identify Coordinates: Always start with a pair. In our example, we have x-values and need to find the corresponding y-values using the given equation.
• Utilize the Axis: The x-axis is horizontal, while the y-axis is vertical. Locate each x-value on the x-axis.
• Mark the Points: For each x, find the y-value using the equation and plot it vertically from the x-axis.

The combination of these points will help you see the shape of the function. In your exercise, by plotting the points for x from -3 to 3, you’ll start forming the graph.

Absolute Value Equation

Understanding absolute value equations is crucial for graphing them accurately. The absolute value of a number, represented by |x|, is the distance from zero, so it's always non-negative.

For the equation given, \( y = |x| - 1 \):

• |x| Calculations: The value of |x| equals x if x is zero or positive, and -x if x is negative.
• Impact of -1: This shifts the entire graph down by 1 unit. The initial absolute value graph touches the y-axis at the origin (0,0), but the shift modifies this to (0,-1).

Understanding these components helps you scribble down the right y-values for your graph.

Piecewise Functions

Absolute value equations often connect to piecewise functions due to their dual nature. This means they have different expressions based on the input.

For our function, \( y = |x| - 1 \), the piecewise formulation is:

• For \( x \geq 0 \): The equation is \( y = x - 1 \).
• For \( x < 0 \): The equation adjusts to \( y = -x - 1 \).

This dual approach reveals why the graph of an absolute value function looks like a "V"—each part stems from different expressions. Recognizing this helps you understand and sketch various pieces of the graph accurately.