Problem 63 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a computer or a graphing calculator Let \(f(x)=\left|x^{3}\right|\). Using the same axes, draw the graphs of \(y=f(x), y=f(3 x)\), and \(y=f(3(x-0.8))\), all on the domain \([-3,3]\)

Step-by-Step Solution

Answer

Draw three graphs: (1) \( y = |x^3| \), (2) \( y = |(3x)^3| \), and (3) \( y = |(3(x-0.8))^3| \).

Step 1: Graph the Function y = f(x)

Plot the graph of the function \( y = \left|x^3\right| \) over the domain \([-3, 3]\). The graph is symmetric with respect to the y-axis and resembles an absolute value function, but it's stretched vertically as it's cubic.

Step 2: Apply Horizontal Compression

For the graph of \( y = f(3x) \), note that the factor of 3 inside the function compresses the graph horizontally by a factor of 3. Every point on the graph of \( y = f(x) \) is moved 1/3 closer to the y-axis. Plot this graph on the same axes.

Step 3: Apply Horizontal Translation and Compression

For \( y = f(3(x-0.8)) \), apply a horizontal translation right by 0.8 followed by a compression by a factor of 3. This means the graph of \( y = f(x) \) is first shifted 0.8 units to the right, then compressed horizontally as described before. Ensure the graph reflects both transformations accurately over the domain \([-3, 3]\).

Key Concepts

Absolute Value Functions

Absolute value functions often come across as a fundamental concept in calculus due to their unique graph shape and properties. They express the non-negative value of a function, showing how far a number is from zero, regardless of direction along the x-axis.

Let's consider the basic function given in the exercise: \(y = |x^3|\). This expression involves an absolute value around a cubic function. Unlike a simple linear absolute value \(y = |x|\), which forms a V-shape, \(y = |x^3|\) provides a symmetric graph around the y-axis while taking the shape of a cubic curve, but always staying above the x-axis due to the absolute value.

Key points to remember about absolute value functions include:

• They always produce non-negative results.
• The function is symmetric, indicating its reflection property along the y-axis especially when graphed.
• When incorporated into other transformations like compression or translation, the absolute value's effect persists.

Horizontal Compression

Horizontal compression is another important transformation concept in graph manipulation. It involves squeezing the graph toward the y-axis, altering the appearance of the function but not the core behavior.

In the equation \(y = f(3x)\), the '3' inside the function signifies the graph experiencing horizontal compression. This happens because the inputs to the function \(f(x)\) need to occur three times faster to maintain the same outputs, shrinking the graph's width by a factor of three.

Important aspects include:

• A factor greater than 1 results in compression; values less than 1 (like 1/2) result in expansion.
• Horizontal compression involves dividing the x-values by the factor, making the function seem tighter along the x-axis.
• Though the x-values change, the y-values remain consistent for every equivalent \(x\). Therefore, the function's height doesn't alter.

Horizontal Translation

Apart from compression, horizontal translation is another crucial graph transformation tool needed for precise function manipulation. It shifts the graph left or right along the x-axis without modifying its shape.

For the expression \(y = f(3(x-0.8))\), this transformation occurs due to the \((x - 0.8)\) term. Here, the graph shifts to the right by 0.8 units. We must consider this translation before applying compression, as specified in the expression.

Key factors about horizontal translation:

• A positive term (like \(- 0.8\)) indicates movement to the right, while a negative term (like \(+ 0.8\)) suggests movement to the left.
• Horizontal translation does not alter the function's form; it simply relocates it along the x-axis.
• This shift makes it easier to position functions as per the need of specific data insights or visual alignment.