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Chapter 9: Problem 12

Graph each absolute value function. \(f(x)=|x-1|\)

Short Answer

Expert verified
Plot the vertex at (1, 0) and create a 'V' shape opening upwards.

Step by step solution

01

- Identify the absolute value function

The given function is \( f(x) = |x-1| \). This is an absolute value function translated horizontally.
02

- Determine the vertex

The vertex of the absolute value function \( f(x) = |x - h| \) is at \( (h, 0) \). For the function \( f(x) = |x-1| \), the vertex is at \( (1,0) \).
03

- Plot the vertex

On the coordinate plane, plot the vertex at the point \( (1, 0) \).
04

- Determine the slope

For \( x \geq 1 \), the function behaves like \( f(x) = x-1 \). For \( x < 1 \), the function behaves like \( f(x) = -(x-1) \). Plot points accordingly.
05

- Plot additional points and draw the graph

Choose points on either side of the vertex to plot. For example, at \( x = 0 \), \( f(0) = |0-1| = 1 \) resulting in point \( (0, 1) \). Also, at \( x = 2 \), \( f(2) = |2-1| = 1 \) resulting in point \( (2, 1) \). Connect these points and the vertex with straight lines, forming a 'V' shape.
06

- Finalize the graph

Ensure that the graph shows the 'V' shape with the vertex at \( (1, 0) \) and opens upwards. The arms of the 'V' should continue infinitely in both directions.

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Key Concepts

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

Vertex
In graphing absolute value functions, identifying the **vertex** is crucial. Think of the vertex as the turning point of the graph. For the function ewline When the function is in the form ewline This means: ewline The vertex is at ewline Once you know the vertex, you have a central point to start plotting the graph.
Coordinate Plane
The **coordinate plane** is the stage where your graph comes to life. It consists of two axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
  1. Choose a scale for the axes that fits your function. This makes points easier to plot and the graph more readable.
  2. Plot the vertex you've identified first. This point is your anchor.
By understanding the layout of the coordinate plane, you can accurately plot points that help visualize your function.
Slope
The **slope** of the absolute value function changes direction at the vertex. For the function ewline This is a simple exercise in understanding slopes for linear functions. If the variable inside the absolute value sign is positive, think of it as: ewline When the variable is negative and inside the absolute value sign, think of it as: This tells you how the line rises or falls as you move away from the vertex.
Plotting Points
After pinpointing the vertex and understanding the slopes, the next step is **plotting points**. Here鈥檚 how you can do this effectively:
  • Start with the vertex
  • Choose additional x-values on either side of the vertex
    • Once you have a few points plotted on the coordinate plane, connecting them seamlessly will reveal the characteristic 'V' shape of the absolute value function. Don't forget that the graph continues infinitely in both directions from the vertex.

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    Most popular questions from this chapter

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