Problem 20

Question

Use transformations of graphs to sketch the graphs of $y_{1}, y_{2},$ and $y_{3}$ by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=3-|x|, \quad y_{2}=3-|3 x|, \quad y_{3}=3-\left|\frac{1}{3} x\right|$$

Step-by-Step Solution

Verified

Answer

The graphs of $y_1$, $y_2$, and $y_3$ are transformations of the basic $|x|$ graph involving reflection, stretching/compression, and vertical translation.

1Step 1: Understand Basic Graph of |x|

Begin by recalling the basic graph of $ y = |x| $, which is a "V" shaped graph with the vertex at (0,0). The graph is symmetrical about the y-axis and has lines of slope 1 and -1.

2Step 2: Graph y1 = 3 - |x|

To sketch $ y_1 = 3 - |x| $, start with the graph of $ y = |x| $ and reflect it over the x-axis to get $ y = -|x| $, then translate this graph 3 units upwards. The vertex is now at (0,3), and the lines slope downwards at -1 and 1 from the vertex.

3Step 3: Graph y2 = 3 - |3x|

For $ y_2 = 3 - |3x| $, perform a horizontal compression by a factor of 3 on the basic $ |x| $ graph to make it $ |3x| $, resulting in a graph that is "steeper". Then, reflect this graph over the x-axis to invert it and translate it up by 3 units. The vertex still remains at (0,3) but the slope of the lines is now -3 and 3.

4Step 4: Graph y3 = 3 - |(1/3)x|

Sketch $ y_3 = 3 - |(1/3)x| $ by stretching the basic $ |x| $ graph horizontally by a factor of 3 to form $ |(1/3)x| $, which flattens the "V". Reflect it over the x-axis, and shift it up by 3 units. The vertex remains at (0,3), and the lines slope at -1/3 and 1/3.

5Step 5: Verify with Graphing Calculator

Enter the equations $ y_1 = 3 - |x| $, $ y_2 = 3 - |3x| $, and $ y_3 = 3 - |(1/3)x| $ in a graphing calculator. Check that the graphs match the expected shape and transformations described above, with appropriate slope and positions.

Key Concepts

Absolute Value FunctionHorizontal CompressionVertical ShiftGraphing Calculator Verification

Absolute Value Function

The absolute value function, denoted by $ |x| $, forms a fundamental building block for understanding graph transformations. This function produces a V-shaped graph, symmetrically reflecting across the y-axis, with its vertex at the origin (0,0). The absolute value captures the notion of distance from the origin on a number line. No matter what input you provide, the output remains non-negative.

The graph consists of two linear segments.
On the right side of the origin, it mirrors the equation $ y = x $.
On the left side, it mirrors $ y = -x $.

Understanding this simple function is crucial when manipulating it with various transformations. In the context of the exercise, these transformations include reflections, compressions, and vertical shifts.

Horizontal Compression

A horizontal compression involves squeezing the graph towards the y-axis. For transformations like $ y = |3x| $, the compression modifies the standard $ |x| $ graph. Specifically, each point on the graph moves closer to the y-axis by a factor related to the value multiplying the x-variable.
When compressing horizontally:

If $ y = |3x| $, it compresses the graph by a factor of 3. The graph appears steeper as it approaches the y-axis more sharply.
For $ y = |(1/3)x| $, it's a stretch by the factor of 3, making it less steep.

This transformation explains the changes in line slopes for equations $ y_2 $ and $ y_3 $ given in the exercise.

Vertical Shift

Vertical shifts involve moving the entire graph up or down along the y-axis. For any function $ y = |x| \, \pm \, b $, where $ b $ is a constant, the graph shifts vertically.

Addition of a positive number shifts the graph upwards.
A negative number shifts it downwards.

In the exercise, all graphs ($ y_1, y_2, y_3 $) undergo a vertical upward shift by 3 units due to the "+3" included in each equation. Consequently, their vertices settle at the point (0,3) rather than the origin.

Graphing Calculator Verification

After sketching transformations manually, verifying your graph using a graphing calculator ensures precision. While drawing by hand offers foundational understanding, technology can confirm the accuracy of plotted lines and transformations. Here's how to leverage your calculator effectively:

Enter each equation into the calculator: $ y_1 = 3 - |x| $, $ y_2 = 3 - |3x| $, $ y_3 = 3 - |(1/3)x| $.
Ensure your calculator is set to the correct viewing window to capture all features of the graphs.
Check if the graph reflects your expected results — similar vertex positions with anticipated slopes.

Using a graphing calculator can solidify your understanding by providing a visual confirmation of your manual efforts.

Problem 20

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