Problem 73 - College Algebra

Question

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 vertically compressed cubic function.

Step 1: Identifying the Toolkit Function

The base or toolkit function that we will consider is the cubic function, which is of the form \( f(x) = x^3 \). This function is known for its characteristic shape, with an inflection point at the origin (0, 0).

Step 2: Understanding the Transformation

The given function \( m(x) = \frac{1}{2} x^{3} \) is a vertical scaling transformation of the toolkit function \( f(x) = x^3 \). This is because the coefficient \( \frac{1}{2} \) in front of \( x^3 \) indicates the graph will be compressed vertically by a factor of \frac{1}{2}.

Step 3: Sketching the Graph

To sketch the graph of \( m(x) = \frac{1}{2} x^{3} \), start by drawing the basic shape of \( x^3 \), which passes through points like (-1, -1), (0, 0), and (1, 1). Then apply the vertical compression: points that were at (0, 0) remain the same, while points like (-1, -1) and (1, 1) will move to (-1, -\frac{1}{2}) and (1, \frac{1}{2}), respectively, thus forming a gentler curve.

Key Concepts

Function Transformation

In mathematics, a transformation changes the position, shape, or size of a function's graph. When dealing with cubic functions, like our toolkit function \( f(x) = x^3 \), transformations can often make the graph stretch, shrink, or shift in various directions. Specifically, the function \( m(x) = \frac{1}{2} x^{3} \) represents a vertical transformation applied to the cubic function.

Here, the transformation involves multiplying the cubic function by \( \frac{1}{2} \). This factor controls how much the graph is vertically compressed or stretched from its original shape. Function transformations fundamentally retain the structure of the original graph. This means that even though the shape changes slightly (due to vertical compression or stretching), the graph still retains its endpoints and the general directionality of the curve.

Vertical Compression

Vertical compression is a type of transformation that effectively 'squeezes' the graph closer to the x-axis, reducing its height. It's like pressing down on a spring but without changing its width.

For our function \( m(x) = \frac{1}{2} x^{3} \), the graph undergoes a vertical compression. This transformation occurs because of the constant \( \frac{1}{2} \) placed in front of \( x^3 \). This means every point on the graph of \( f(x) = x^3 \) is quantitatively halved in the vertical direction.

• The original point \( (1, 1) \) on the basic cubic graph becomes \( (1, \frac{1}{2}) \).
• Similarly, \( (-1, -1) \) transposes to \( (-1, -\frac{1}{2}) \).

Vertical compression doesn’t affect points located on the x-axis, like \( (0, 0) \), since zero multiplied by any number remains zero. This results in a graph with a "gentler" curve which maintains the same x-intercepts while decreasing the steepness of slopes.

Graph Sketching

Sketching graphs with transformations involves understanding both the original function shape and the effects of modifications applied to it. For a function like \( m(x)=\frac{1}{2} x^{3} \), you start by sketching the basic shape of \( x^3 \). This graph has a distinctive ‘S’ shape, passing through key points such as \( (-1,-1) \), \( (0,0) \), and \( (1,1) \).

• Begin by plotting the basic cubic curve for the toolkit function \( f(x) = x^3 \).
• Apply the vertical compression: halve the y-value of each plotted point.

For instance, adjust the point \( (1,1) \) to \( (1, \frac{1}{2}) \) to reflect the new value. Repeat this for all key points.

The new graph is a less steep curve, maintaining the original points' symmetry but with a condensed vertical span. By visualizing the transformation process, one can ensure that key characteristics of the graph align appropriately, aiding in accurate and intuitive graph sketching.