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Chapter 3: Problem 24

Graph each function. Be sure to label three points on the graph. $$f(x)=|x|$$

Short Answer

Expert verified
Plot points (-1,1), (0,0), and (1,1), then connect them to form a V shape.

Step by step solution

01

- Identify Key Points

Identify three key points to plot for the function. For the absolute value function, let's choose x-values: -1, 0, and 1.
02

- Evaluate the Function at Key Points

Calculate the value of the function at each of the chosen x-values. \ For x = -1: \ \(f(-1) = |-1| = 1\) \ For x = 0: \ \(f(0) = |0| = 0\) \ For x = 1: \ \(f(1) = |1| = 1\)
03

- Plot the Points

Plot the points on a coordinate plane: (-1,1), (0,0), and (1,1).
04

- Sketch the Graph

Connect the plotted points with straight lines to form the V-shape characteristic of the absolute value function. The lines should form a 'V' shape centered at the origin (0,0).
05

- Label the Points

Clearly label the three points you plotted: (-1,1), (0,0), and (1,1) on the graph. Ensure the x-axis and y-axis are also labeled.

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Key Concepts

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

Coordinate Plane
Understanding the coordinate plane is the first step to graphing any function. The coordinate plane is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
Each point on the plane can be described by an ordered pair \(x, y\), where \x\ represents the horizontal distance from the origin and \y\ represents the vertical distance. The origin is the point \(0,0\) where both axes intersect.

Here are some important aspects:
  • The horizontal (x) values increase to the right and decrease to the left of the origin.
  • The vertical (y) values increase when moving up and decrease when moving down from the origin.

When plotting points, it's crucial to start at the origin and move according to the ordered pair's values. This understanding lays the foundation for accurately plotting any function.
Plotting Points
Plotting points involves placing points on the coordinate plane according to their coordinates. Let's use the example from the exercise to understand this process. We have the points (-1, 1), (0, 0), and (1, 1).
  • First point: (-1, 1). Start at the origin, move 1 unit left, and then 1 unit up.
  • Second point: (0, 0). This is the origin, so no movement is needed.
  • Third point: (1, 1). Start at the origin, move 1 unit right, and then 1 unit up.

Plotting these points is crucial for graphing the absolute value function efficiently. Once plotted correctly, connections between them will visualize the function.
V-shape Graph
The absolute value function \(f(x) = |x|\) creates a distinct V-shape graph. This shape occurs because the function outputs the distance of any x-value from zero, effectively flipping negative values to positive. Here's a closer look at the steps to sketch this graph:
  • Identify and plot points: Use key points like (-1, 1), (0, 0), and (1, 1).
  • Draw the lines: Connect these points with straight lines.
    This will naturally form a V-shape.
  • Label your points: Ensure the x-axis and y-axis, as well as the points, are clearly labeled.

This V-shape is symmetric around the y-axis and centered at the origin. Understanding this will help in accurately sketching and interpreting absolute value functions.

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Most popular questions from this chapter

A ball is thrown upward from the top of a building. Its height \(h,\) in feet, after \(t\) seconds is given by the equation \(h=-16 t^{2}+96 t+200 .\) How long will it take for the ball to be \(88 \mathrm{ft}\) above the ground?

What is the conjugate of \(\frac{3}{2}-2 i ?\) [This problem is based on content from Section 1.3 , which is optional. \(]\)

Use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. \(f(x)=-0.4 x^{3}+0.6 x^{2}+3 x-2 \quad[-4,5]\)

The period \(T\) (in seconds) of a simple pendulum is a function of its length \(l\) (in feet) defined by the equation $$ T=2 \pi \sqrt{\frac{l}{g}} $$ where \(g \approx 32.2\) feet per second per second is the acceleration due to gravity. (a) Use a graphing utility to graph the function \(T=T(l)\). (b) Now graph the functions \(T=T(l+1), T=T(l+2)\) $$ \text { and } T=T(l+3) $$ (c) Discuss how adding to the length \(l\) changes the period \(T\) (d) Now graph the functions \(T=T(2 l), T=T(3 l)\), and \(T=T(4 l)\) (e) Discuss how multiplying the length \(l\) by factors of 2,3 , and 4 changes the period \(T\)

Determine algebraically whether each function is even, odd, or neither. \(h(x)=\frac{x}{x^{2}-1}\)
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