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Chapter 2: Problem 66

Use transformations to graph the functions. $$ q(x)=\frac{1}{3}|x+2|-1 $$

Short Answer

Expert verified
Graph of \( q(x) = \frac{1}{3}|x+2| -1 \) is the absolute value function shifted left 2 units, compressed vertically by \( \frac{1}{3} \), and shifted down 1 unit.

Step by step solution

01

Identify the Parent Function

The parent function for the given equation is the absolute value function, which is written as: \[ f(x) = |x| \].
02

Horizontal Shift

Examine the expression within the absolute value, \( |x+2| \). This indicates a horizontal shift. The function is shifted 2 units to the left. Apply the transformation to the parent function: \[ f(x) = |x+2| \].
03

Vertical Stretch/Compression

Notice the coefficient \( \frac{1}{3} \) in front of the absolute value. This represents a vertical compression by a factor of \( \frac{1}{3} \). Apply this transformation: \[ f(x) = \frac{1}{3}|x+2| \].
04

Vertical Shift

Finally, observe the \( -1 \) outside the absolute value, indicating a vertical shift. This means the graph will shift 1 unit down. Apply the transformation: \[ q(x) = \frac{1}{3}|x+2| -1 \].
05

Plot the Graph

Using the transformations identified, you can now plot the graph on a coordinate plane. Start with the vertex at (-2, -1) due to shifts and include a vertical compression factor, ensuring the slopes of the lines are \( \frac{1}{3} \).

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Key Concepts

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

absolute value function
The absolute value function is a fundamental concept in Algebra. The basic form of this function is written as \( f(x) = |x| \). It creates a V-shaped graph that opens upwards, and its vertex, which is the lowest point, is at the origin (0,0). The absolute value of any number is always positive, reflecting the distance of that number from zero on the number line. Because of this, the graph mirrors about the y-axis. Understanding the absolute value function is crucial before adding transformations like shifts or compressions.
horizontal shift
A horizontal shift moves the graph of a function left or right on the coordinate plane. This type of transformation occurs within the expression inside the absolute value. For instance, \( |x+2|\) translates the graph of \( |x| \) 2 units to the left. It may seem counterintuitive, but adding a positive number inside the function results in a leftward shift. To remember this better: \(f(x) = |x-h|\) moves \(h\) units horizontally in the opposite direction of the sign you see inside.
vertical compression
Vertical compression changes the steepness of the graph of a function. For the absolute value function, introducing a coefficient less than 1 but greater than 0 outside the absolute value creates this effect. In \( q(x) = \frac{1}{3}|x+2|\), the graph is compressed vertically by a factor of \(\frac{1}{3}\). This means that it becomes less steep than the original \(|x|\) and appears 'wider' apart from the center. This change maintains the V-shape but alters the slopes.
vertical shift
Vertical shifts adjust the position of the graph up or down on the coordinate plane. This modification is made by adding or subtracting a constant after the absolute value term. In the expression \( q(x) = \frac{1}{3}|x+2| - 1\), the graph is shifted 1 unit down. Vertical shifts are easier to spot since they modify the value of the function directly. This kind of shift keeps the overall shape intact but changes the location of the vertex. The lowest point of the vertex here moves to \( (-2, -1) \).

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Most popular questions from this chapter

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