Problem 75 - College Algebra

Question

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ p(x)=\left(\frac{1}{3} x\right)^{3}-3 $$

Step-by-Step Solution

Answer

The graph is a horizontally stretched (factor of 3) and vertically shifted down by 3 units version of \( x^3 \).

Step 1: Identify the Toolkit Function

The given function is \( p(x) = \left( \frac{1}{3} x \right)^{3} - 3 \). The basic toolkit function here is the cube function \( f(x) = x^3 \). Our task is to identify how the given function is a transformation of this basic function.

Step 2: Apply Horizontal Transformation

The expression \( \left( \frac{1}{3} x \right)^3 \) indicates a horizontal stretch. Since \( x \) is multiplied by \( \frac{1}{3} \), it implies that the graph is stretched horizontally by a factor of 3. This means each point on the transformed graph is three times further from the y-axis compared to the base function \( f(x) = x^3 \).

Step 3: Apply Vertical Transformation

The subtraction \(-3\) in the expression \( \left( \frac{1}{3} x \right)^{3} - 3 \) indicates a vertical shift. The entire graph is moved down 3 units. This means that the y-coordinate of every point on the graph of \( \left( \frac{1}{3} x \right)^3 \) is reduced by 3.

Step 4: Sketch the Graph

To sketch the graph, start by drawing the basic cubic graph \( f(x) = x^3 \). Apply the horizontal stretch, making sure that it is 3 times wider. Then shift the entire graph downward by 3 units. The transformed graph should look similar to a wider version of the cubic graph positioned lower on the y-axis.

Key Concepts

Cubic Function

The cubic function, denoted as \( f(x) = x^3 \), is one of the basic toolkit functions in mathematics. It's often represented as the parent graph for any function involving an \( x^3 \) term. Recognizing the properties of this fundamental graph is key:

• The graph of \( x^3 \) is symmetric with respect to the origin, meaning it has rotational symmetry when turned 180 degrees.
• This graph passes through the origin point (0,0), serving as a reference point for transformations.
• The curve approaches zero more steeply than a linear function but less steeply than a quadratic function.

Understanding how transformations affect the cubic function enables us to analyze more complex forms. Once the basic features of \( x^3 \) are recognized, subsequent transformations like stretches or shifts become more intuitive to visualize and draw.

Horizontal Stretch

A horizontal stretch changes the width of the graph. Mathematically, this is displayed in the term \( \left( \frac{1}{3} x \right)^3 \), where the \( x \) is multiplied by \( \frac{1}{3} \). Here are a few essential points:

• Since \( x \) is replaced by \( \frac{1}{3}x \), each point on the graph is moved farther from the y-axis by a factor of 3. Hence, the graph is three times as wide.
• This manipulation makes the graph appear stretched horizontally, emphasizing its relative width compared to the basic cubic graph \( f(x) = x^3 \).
• Understanding horizontal transformations involves recognizing how adjustments in the \( x \)-input affect the graph's shape across the y-axis.

Visualizing a horizontal stretch helps when sketching the new, wider graph, emphasizing the increased distance of points from the y-axis.

Vertical Shift

A vertical shift modifies the graph's position along the y-axis. This transformation is indicated by the \(-3\) in the formula \( \left( \frac{1}{3} x \right)^3 - 3 \). Here's how it works:

• A vertical shift affects the entire graph uniformly, moving every point down 3 units due to the \(-3\).
• This alteration mirrors the change in the value of each \( y \)-output by subtracting 3, effectively lowering the entire function on the graph.
• Vertical shifts are straightforward, adjusting the location of the curve without altering its shape. Instead, it modifies where the curve sits in relation to the x-axis.

Visualizing vertical shifts aids in creating accurate sketches, as it focuses on re-positioning the function's graph lower on the coordinate plane by counting down units.