Problem 13

Question

Use the graph of $y=|x|$ and information from this section (but not a calculator) to sketch the graph of the function. $$g(x)=|x|-2$$

Step-by-Step Solution

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Answer

Answer: The vertex of the graph of the function $g(x)=|x|-2$ is at the point (0, -2).

1Step 1: Understand the given function

The given function is $g(x)=|x|-2$. This function is related to the original function $y=|x|$ by a vertical translation. The -2 term indicates that the graph of the function $g(x)$ is obtained by shifting the graph of $y=|x|$ two units downward.

2Step 2: Sketch the graph of $y=|x|$

Begin by sketching the graph of the function $y=|x|$. This is a simple V-shaped graph that starts from the origin (0,0) and moves upwards to the right and left with a slope of 1 and -1, respectively. The graph consists of two line segments: one moving to the right upwards and the other moving to the left upwards, both making an angle of 45 degrees with the x-axis.

3Step 3: Perform the vertical translation

In this step, we will perform the vertical translation by shifting the graph of the original function $y=|x|$ two units downward. To do this, move each point of the graph of $y=|x|$ two units down along the vertical axis (y-axis). In particular, the vertex of the graph of $y=|x|$ at the origin (0,0) will move to the point (0, -2).

4Step 4: Sketch the new graph

Now that we have performed the vertical translation, it's time to sketch the graph of the new function $g(x)=|x|-2$. The graph still has the V-shape, but its vertex is now located at the point (0, -2). Draw the two line segments of the graph, still having slopes of 1 and -1, respectively. The graph now starts from the point (0, -2) and moves upwards to the right and left. The resulting graph represents the function $g(x)=|x|-2$.

Key Concepts

Absolute Value FunctionVertical TranslationFunction SketchingMathematical Graphs

Absolute Value Function

Let's start with understanding the absolute value function, which is denoted as $ y = |x| $. The absolute value of a number represents its distance from zero on a number line, always yielding a non-negative result. For instance, $ |3| = 3 $ and $ |-3| = 3 $. When you graph $ y = |x| $, you get a distinct V-shape.

The graph has a vertex at the origin (0,0).
It moves upwards to the right (for positive values of $x$) and to the left (for negative values of $x$).
This results in line segments with slopes of 1 and -1.

This symmetry and V-shape make the absolute value graph quite straightforward to spot and sketch.

Vertical Translation

Vertical translation is a simple transformation that shifts a graph up or down. To perform a vertical translation, you add or subtract a constant from the function. For example, given the function $ y = |x| $ and the function $ g(x) = |x| - 2 $, you're instructed to shift every point of $ y = |x| $ downward by 2 units. This is because of the $-2$.

The vertex of the original graph $ y = |x| $ at (0,0) moves to (0,-2).
The rest of the graph follows suit, preserving the shape and angles.

This transformation maintains the same shape but changes its position. It's a handy method to adjust where a graph sits on the coordinate plane without altering its overall form.

Function Sketching

Sketching a function involves visually representing it with a graph to understand its behavior. Here's how you sketch the $ g(x) = |x| - 2 $ function:

Start with the basic graph $ y = |x| $, which is a V-shape centered at the origin.
Apply the vertical translation by shifting the entire graph down by 2 units.
Draw the vertex of the new function at (0, -2) instead of (0, 0).
Maintain the same slopes of 1 and -1 for the arms of the V.

By following these steps, you can accurately sketch $ g(x) = |x| - 2 $ or similar functions by focusing on transformations.

Mathematical Graphs

Mathematical graphs are visual representations of equations and functions. Understanding how to read and sketch them is fundamental to visualizing how functions behave. Graphs convey information such as:

The general behavior and symmetry of a function.
Specific points, like intercepts, vertices, and asymptotes.
Changes in position through transformations, such as translations, reflections, or stretches.

In this context, starting with $ y = |x| $ gives you a basic framework to build on. Then, understanding transformations allows you to modify and understand its new behavior. This practice is key to mastering graph transformations and applying them to various functions.

Problem 13

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