Problem 44 - Algebra for College Students

Question

(a) Graph \(f(x)=|x|, f(x)=2|x|, f(x)=4|x|\), and \(f(x)=\frac{1}{2}|x|\) on the same set of axes. (b) Graph \(f(x)=|x|, f(x)=-|x|, f(x)=-3|x|\), and \(f(x)=-\frac{1}{2}|x|\) on the same set of axes. (c) Use your results from parts (a) and (b) to make a conjecture about the graphs of \(f(x)=a|x|\), where \(a\) is a nonzero real number. (d) Graph \(f(x)=|x|, f(x)=|x|+3, f(x)=|x|-4\), and \(f(x)=|x|+1\) on the same set of axes. Make a conjecture about the graphs of \(f(x)=|x|+k\), where \(k\) is a nonzero real number. (e) Graph \(f(x)=|x|, f(x)=|x-3|, f(x)=|x-1|\), and \(f(x)=|x+4|\) on the same set of axes. Make a conjecture about the graphs of \(f(x)=|x-h|\), where \(h\) is a nonzero real number. (f) On the basis of your results from parts (a) through (e), sketch each of the following graphs. Then use a graphing calculator to check your sketches. (1) \(f(x)=|x-2|+3\) (2) \(f(x)=|x+1|-4\) (3) \(f(x)=2|x-4|-1\) (4) \(f(x)=-3|x+2|+4\) (5) \(f(x)=-\frac{1}{2}|x-3|-2\)

Step-by-Step Solution

Answer

\(f(x) = a|x|\) stretches/compresses and reflects based on \(a\); \(f(x) = |x| + k\) shifts vertically; \(f(x) = |x-h|\) shifts horizontally.

Step 1: Understand Graph Transformations

To graph functions involving absolute values such as \(f(x) = a|x|\), we need to understand that \(a\) affects the vertical stretch or compression of the graph. If \(a > 1\), it stretches the graph vertically; if \(0 < a < 1\), it compresses it. If \(a < 0\), it reflects the graph across the x-axis.

Step 2: Graph (a) Absolute Value Stretches and Compressions

Start with the base graph of \(f(x) = |x|\), which is a V-shape centered at the origin. For \(f(x) = 2|x|\), the graph opens wider, as the slope increases to 2. \(f(x) = 4|x|\) opens even wider. For \(f(x) = \frac{1}{2}|x|\), the graph is compressed, making it narrower.

Step 3: Graph (b) Negative Reflections

For this part, the graphs involve negative coefficients. Start with \(f(x) = |x|\). Reflect it to get \(f(x) = -|x|\). Then, \(f(x) = -3|x|\) is a vertically stretched reflection, and \(f(x) = -\frac{1}{2}|x|\) is a compressed reflection.

Step 4: Conjecture Based on (a) and (b)

From parts (a) and (b), we can make a conjecture that \(f(x) = a|x|\) results in a vertical stretch or compression based on \(|a|\) and a reflection if \(a < 0\). The graph is always a V-shape centered at the origin.

Step 5: Graph (d) Vertical Translations

For \(f(x) = |x| + k\), start with \(f(x) = |x|\). Adding 3 gives \(f(x) = |x| + 3\), which shifts the graph up by 3 units. Subtracting 4 gives \(f(x) = |x| - 4\), shifting it down by 4 units. \(f(x) = |x| + 1\) shifts up by 1 unit.

Step 6: Conjecture Based on (d)

The graphs \(f(x) = |x| + k\) are vertical translations of the base graph \(f(x) = |x|\). The value of \(k\) moves the graph up (if positive) or down (if negative).

Step 7: Graph (e) Horizontal Translations

For \(f(x) = |x - h|\), use the base graph of \(f(x) = |x|\). \(f(x) = |x-3|\) moves the graph 3 units to the right. \(f(x) = |x-1|\) moves it 1 unit to the right. \(f(x) = |x+4|\) moves it 4 units to the left.

Step 8: Conjecture Based on (e)

Graphs of \(f(x) = |x-h|\) are horizontal translations of \(f(x) = |x|\). The graph moves to the right if \(h\) is positive and to the left if \(h\) is negative.

Step 9: Sketch and Verify Graphs in (f)

Using transformations learned, sketch the graphs: (1) \(f(x) = |x-2| + 3\) from (e) and (d), (2) \(f(x) = |x+1| - 4\), (3) \(f(x) = 2|x-4| - 1\), (4) \(f(x) = -3|x+2| + 4\), (5) \(f(x) = -\frac{1}{2}|x-3| - 2\). Verify using a graphing calculator.

Key Concepts

Graph Transformations

When dealing with absolute value functions, understanding graph transformations is vital. These transformations are changes made to the base graph of the function, typically represented as \( f(x) = |x| \). Each transformation involves adjusting parts of this V-shaped graph.

• The coefficient \( a \) in \( f(x) = a|x| \) determines the vertical stretch or compression. If \(|a| > 1\), the graph stretches. If \(0 < |a| < 1\), it compresses.
• Negative values of \( a \) reflect the graph across the x-axis.

These transformations keep the graph's general shape but change how it interacts with the axes.

Vertical Translations

Vertical translations adjust the graph's position up or down on the y-axis. This is influenced by the constant \( k \) in the function \( f(x) = |x| + k \). Here’s how it works:

• If \( k > 0 \), the graph shifts upwards by \( k \) units.
• If \( k < 0 \), it shifts downwards by \( |k| \) units.

This type of transformation doesn't affect the shape of the V, only its vertical placement. It’s beneficial for quickly evaluating shifts by merely examining the function.

Horizontal Translations

Horizontal translations involve moving the graph sideways along the x-axis. These are controlled by the parameter \( h \) in the equation \( f(x) = |x - h| \).

• If \( h > 0 \), the graph moves to the right by \( h \) units.
• If \( h < 0 \), it moves to the left by \( |h| \) units.

Horizontal translations shift every point of the graph left or right, making them essential to fine-tune the position without altering the graph’s overall layout.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing function transformations. It helps confirm that the theoretical manipulations of a function's equation match the expected changes on the graph.

• Input the function to see the graph's accurate shape and position.
• Easily perform comparisons between different translated or transformed graphs.
• Adjust variables such as \( a \), \( h \), and \( k \) to see real-time transformations.

By using a graphing calculator, students can practice sketching by hand and then verify their graphs, enhancing learning and accuracy.