Problem 16

Question

a. The graph of $f(x)=|x|-5$ is the same as the graph of $f(x)=|x|$ except that it is shifted _____ units _____. b. The graph of $f(x)=|x-5|$ is the same as the graph of $f(x)=|x|$ except that it is shifted _____ units to the _____.

Step-by-Step Solution

Verified

Answer

a. 5 units downward; b. 5 units to the right.

1Step 1: Understanding Vertical Shifts

Given the function $ f(x) = |x| - 5 $, this represents a vertical shift of the graph $ f(x) = |x| $. When a constant $ c $ is subtracted from a function, $ f(x) = |x| - c $, the graph shifts downward by $ c $ units. Thus, the graph of $ f(x) = |x| - 5 $ is shifted 5 units downward.

2Step 2: Understanding Horizontal Shifts

The function $ f(x) = |x-5| $ represents a horizontal shift of the graph $ f(x) = |x| $. When a constant $ h $ is subtracted from $ x $, $ f(x) = |x-h| $, the graph shifts to the right by $ h $ units. Therefore, the graph of $ f(x) = |x-5| $ is shifted 5 units to the right.

Key Concepts

Vertical ShiftsHorizontal ShiftsAbsolute Value Functions

Vertical Shifts

When dealing with graph transformations, vertical shifts are changes that move the graph up or down on the coordinate plane. The basic concept is straightforward:

If you have a function like $ f(x) = |x| $ and you subtract a constant $ c $, forming $ f(x) - c $, the entire graph shifts downward by $ c $ units. This is known as a vertical shift.
Conversely, if you add a constant $ c $, forming $ f(x) + c $, the graph moves upward by $ c $ units.

A vertical shift does not affect the shape of the graph, only its position along the vertical axis. For instance, the function $ f(x) = |x| - 5 $ means you take the graph of $ f(x) = |x| $ and move it 5 units downward. This shift results from the subtraction of 5 in the function, indicating a downward translation along the y-axis.

Horizontal Shifts

Horizontal shifts may seem slightly trickier because they involve changes inside the argument of the function. Essentially, they move the graph left or right. Let's break it down:

For a function like $ f(x) = |x-h| $, the graph shifts to the right by $ h $ units if $ h $ is positive.
If you subtract a negative value $-h$ (for example $ f(x) = |x-(-h)| $ or $ f(x) = |x+h| $), the graph moves to the left by $ h $ units.

The important takeaway is that the direction of the shift is opposite to the sign of $ h $ inside the function. For instance, in the function $ f(x) = |x - 5| $, you subtract 5 directly from $ x $, meaning the graph is shifted 5 units to the right. This is a common place where students might confuse the direction, so remember: subtracting a positive number moves the graph to the right.

Absolute Value Functions

Absolute value functions have a unique characteristic in how they shape graphs. The absolute value function is represented as $ f(x) = |x| $. What makes it special?

The function $ |x| $ equals $ x $ when $ x $ is positive or zero, and $ -x $ when $ x $ is negative.
This results in a V-shaped graph that has its vertex at the origin (0,0).

Absolute value functions are often transformed by shifting or stretching them, but the V-shape remains. A classic transformation is combining absolute values with vertical and horizontal shifts, like in $ f(x) = |x| - 5 $ or $ f(x) = |x-5| $. These transformations change the position and orientation of the V-shape without altering its fundamental form. Recognizing the vertex and understanding shifts helps greatly when graphing these functions.

Problem 16

Other exercises in this chapter

Problem 16

Complete each factorization. $$ \begin{aligned} x^{3}-x^{2}+2 x-2 &=(x-1)+(x-1) \\ &=(\quad)\left(x^{2}+2\right) \end{aligned} $$

View solution

Problem 16

Factor. $144 x^{2}-1$

View solution

Problem 16

Write the inequality $10>|16 x-3|$ in an equivalent form with the absolute value expression on the left side.

View solution

Problem 16

a. Graph: $(-\infty, 2) \cup[3, \infty)$ b. Graph: $(-\infty, 3) \cap[-2, \infty)$

View solution