Problem 73 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. \(m(x)=\frac{1}{2} x^{3}\)

Step-by-Step Solution

Answer

The function is a vertical compression of \( x^3 \) by a factor of \( \frac{1}{2} \).

Step 1: Identify the Toolkit Function

The given function is a transformation of the toolkit function for a cubic function, which is typically denoted as \( f(x) = x^3 \). This function represents the simplest form of a cubic graph.

Step 2: Identify the Transformation

The function \( m(x) = \frac{1}{2}x^3 \) is a vertical compression of the toolkit function \( f(x) = x^3 \). The coefficient \( \frac{1}{2} \) in front of the \( x^3 \) term compresses the graph vertically by a factor of \( \frac{1}{2} \).

Step 3: Sketch the Original Toolkit Function

Begin by sketching the graph of \( f(x) = x^3 \), which is a basic cubic curve passing through the origin (0,0). It has a symmetric S-shape, increasing in the positive x-direction and decreasing in the negative x-direction.

Step 4: Sketch the Transformed Function

Transform the graph of \( f(x) = x^3 \) to \( m(x) = \frac{1}{2}x^3 \) by compressing it vertically. For each point \((x, y)\) on \( f(x) = x^3 \), plot the point \((x, \frac{1}{2}y)\) on the new graph. The graph remains symmetric and passes through the origin, but rises and falls less steeply.

Key Concepts

Vertical Compression

Vertical compression occurs when a function's graph is "squished" towards the x-axis, making it appear less steep. In mathematical terms, this happens when the factor in front of the function is a fraction, like \( \frac{1}{2} \). If the original toolkit function is \( f(x) = x^3 \), applying the transformation \( m(x) = \frac{1}{2}x^3 \) compresses the graph vertically. This means each output value at \( y \) is halved, making the curve less steep while maintaining its basic shape and symmetry.

**Key Characteristics of Vertical Compression:**

• The key points remain on the line but are closer to the x-axis.
• The graph maintains its shape; it still passes through the origin.
• The transformation affects how quickly the graph rises or falls.

In terms of visualization, imagine the original graph being pushed down gently from the top, so it touches the x-axis sooner.

Toolkit Function

In algebra, a toolkit function represents the basic or simplest version of a type of function. For cubic functions, the toolkit function is \( f(x) = x^3 \). It is vital because it establishes a base shape that other cubic functions transform from.

**Characteristics of the Cubic Toolkit Function:**

• It passes through the origin (0,0).
• The graph is an S-shaped curve that is symmetric about the origin.
• It increases infinitely as x becomes positive and decreases infinitely as x becomes negative.

Understanding the cubic toolkit function is essential because any transformations, like compressions or stretches, are directly compared to this shape. All parts of the curve, each point and its distance from the x-axis, set the stage for how changes will appear in transformed versions.

Graph Transformations

Graph transformations involve altering a function's graph through various modifications like shifting, reflecting, stretching, or compressing. These transformations help understand how a base graph can change to meet specific characteristics.

**Common Types of Graph Transformations:**

• Vertical Stretch/Compression: Changes the steepness by multiplying/dividing the output values.
• Horizontal Shift: Moves the graph left or right on the plane.
• Vertical Shift: Lifts or lowers the graph along the y-axis.
• Reflection: Flips the graph across the x-axis or y-axis.

In the case of \( m(x) = \frac{1}{2}x^3 \), the transformation is a vertical compression. Each transformation is a step in modifying the toolkit function's graph to suit a new purpose while keeping core characteristics intact, like symmetry or the origin's position as that's fundamental to cubic functions.