Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.$$\begin{aligned}&f(x)=|x|\\\&g(x)=|2 x|\\\&h(x)=-2|x+2|-1\end{aligned}$$.

Step-by-Step Solution

Answer

The sketch will consist of three 'v' shaped lines on a graph. The first one is a standard 'v' shape passing through the origin, the second one is a narrower 'v' shaped line due to the coefficient of 2 and the last one is an inverted 'v' shape which is shifted to the left by 2 units and downward by 1 unit.

Step 1 - Graph of Function \(f(x) = |x|\)

The graph of \(f(x) = |x|\) is a straight line passing through the origin, forming a 'v' shape. Positive x-values result in positive y-values and vice versa. In other words, as x increases, y increases, and as x decreases, y still increases.

Step 2 - Graph of Function \(g(x) = |2x|\)

The graph of \(g(x) = |2x|\) is like that of \(f(x) = |x|\), except it becomes steeper due to the coefficient of 2. The 'v' shape will be narrower.

Step 3 - Graph of Function \(h(x) = -2|x + 2| - 1\)

The graph of \(h(x) = -2|x + 2| - 1\) is similar to the absolute function but it's inverted due to the negative sign in front of the 2. The graph shifts to the left by 2 units due to \(x + 2\) in the absolute value. Further, the entire graph shifts downward by 1 unit due to the \(- 1\) at the end of the function.

Key Concepts

Graph Transformations

Understanding how graph transformations work is essential when dealing with absolute value functions. A graph transformation alters the basic graph of a function in one or more ways. These transformations can include shifts, stretches, compressions, and reflections.

There are a few transformations to consider for absolute value functions like

• Shifts: Horizontal or vertical movement of the graph. For example, in the function \( h(x) = -2|x+2| -1 \), the graph shifts left by 2 units due to \( x+2 \) and down by 1 due to the -1.
• Stretches and Compressions: Changing the graph's "steepness" or "width." A coefficient greater than 1 like in \( g(x)=|2x| \) compresses the graph, making it steeper.
• Reflections: Flipping the graph over an axis. The function \( h(x) = -2|x+2| - 1 \) reflects the graph over the x-axis due to the negative sign.

By learning these transformations, one can easily plot and visualize different forms of a basic function quickly and accurately.

Piecewise Functions

Piecewise functions divide the function into separate parts, each with its own rule or expression. This means different formulas apply to different sections of the domain. The absolute value function \( f(x) = |x| \) is a common example of a piecewise function. Depending on the value of \( x \), it defines different rules:

• For \( x \geq 0 \): \( f(x) = x \)
• For \( x < 0 \): \( f(x) = -x \)

This ensures that the outputs are always non-negative, translating geometrically to the characteristic 'V' shape seen in its graph. When dealing with transformations, it is important to understand how each part of the piecewise function undergoes changes. For example, in \( h(x) = -2|x + 2| - 1 \), these changes apply differently to each segment of the domain, influencing the overall shape of the graph.

Graphing Utilities

Graphing utilities are digital tools that help visualize mathematical functions easily. They can verify manual sketches by allowing students to enter function equations and see the precise curve plotted on a coordinate plane.

Using graphing software or calculators, you can quickly graph and compare different transformations, such as in the original exercise with the functions \( f(x) = |x|, g(x) = |2x|, \) and \( h(x) = -2|x + 2| - 1 \). These tools allow users to:

• Check the accuracy of hand-drawn graphs efficiently.
• Zoom in and out to observe specific aspects of the graph.
• Experiment with different constants and coefficients to see real-time changes.
• Overlay multiple graphs to compare their transformations directly.

Graphing utilities are particularly useful for students seeking to strengthen their understanding of mathematical concepts through immediate and accurate visual feedback.