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Chapter 1: Problem 24

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \(x = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檒l 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"鈥攅ach part stems from different expressions. Recognizing this helps you understand and sketch various pieces of the graph accurately.

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Most popular questions from this chapter

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{x}$$

If you are given a function's graph, how do you determine if the function is even, odd, or neither?

I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

Use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+y^{2}=25$$

Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$\begin{aligned}x^{2}+y^{2} &=16 \\\x-y &=4\end{aligned}$$
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