Problem 105 - College Algebra

Question

Begin by graphing the standard cubic function, \(f(x)-x^{3} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac 12(x-3)^{3}-2 $$

Step-by-Step Solution

Answer

To transform the graph of \(f(x) = x^{3}\) into \(h(x) = \frac{1}{2}(x-3)^{3}-2\), you need to shift it to the right by 3 units, scale it down vertically by a factor of \(\frac{1}{2}\), and then shift it downwards by 2 units.

Step 1: Graph the standard cubic function

Start by drawing the graph of \(f(x) = x^{3}\). The graph of a cubic function is always symmetric with respect to the origin and continues to rise or fall as \(x\) increases or decreases. At \(x=0\), \(y=0\). As \(x\) increases, \(y\) also increases, albeit at a faster rate. As \(x\) decreases, \(y\) also decreases, also at a faster rate.

Step 2: Identify the transformation parameters

In the function \(h(x) = \frac{1}{2}(x-3)^{3}-2\), there are two transformation parameters in comparison to the standard cubic function. The parameter \(-3\) inside the brackets with \(x\) corresponds to a shift to the right by 3 units. The parameter \(-2\) outside the brackets corresponds to a shift downwards by 2 units. The coefficient \(\frac{1}{2}\) corresponds to a vertical scaling (the graph is 'squished' by a factor of \(\frac{1}{2}\).

Step 3: Graph the function

Begin with the original graph of \(f(x) = x^{3}\). Apply the transformations in the order they were identified: 1) Shift the entire graph right by three units. 2) Then, vertically 'squish' the graph by a factor of \(\frac{1}{2}\). 3) Lastly, shift the entire graph down by two units. You have now drawn the graph of \(h(x) = \frac{1}{2}(x-3)^{3}-2\).

Key Concepts

Graphing Transformations

When you transform a graph, you are essentially altering its shape, position, or orientation on the coordinate plane. Transformations make it possible to modify functions while maintaining their core characteristics.
Let's discuss a few key types of transformations:

• Vertical transformations: These include shifts up or down and vertical scaling. Shifting changes the position, while scaling changes the steepness of the graph.
• Horizontal transformations: Shifts left or right and horizontal scaling fall under this category. Horizontal shifts alter the graph's position left or right, while horizontal scaling changes its width.
• Reflections: These occur when a graph is flipped over a line, such as the x-axis or y-axis.

In the given exercise, the cubic function is transformed using a horizontal shift right, vertical scaling, and a vertical downward shift. To successfully transform a graph, follow a step-by-step approach of identifying, understanding, and applying each transformation in succession. This method leads to an accurate depiction on the graph.

Cubic Functions

Cubic functions are polynomial functions of degree three and take the general form of \( f(x) = ax^3 + bx^2 + cx + d \). These functions can create a variety of shapes because of their versatile nature. The most basic cubic function is \( f(x) = x^3 \).
Some core characteristics of cubic functions include:

• Symmetry: They are symmetric with respect to the origin if we only stick to the basic form \( x^3 \), meaning if you rotate the graph 180 degrees around the origin, it'll look the same.
• End behavior: As \( x \) gets large in either the positive or negative direction, the function rises or falls steeply.
• Inflection point: The graph has an inflection point where the curvature changes direction, usually at the origin for \( x^3 \).

In this exercise, understanding the symmetry and inflection point of the basic cubic function helps perform and visualize the transformations accurately.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are two fundamental types of transformations that change the position of the graph without affecting its shape.
Horizontal Shifts:
A horizontal shift moves the graph left or right. This shift is dictated by changes inside the function's argument \( (x - h) \). If \( h \) is positive, the graph shifts \( h \) units to the right; if \( h \) is negative, the shift is \(|h|\) units to the left. In the exercise function \( h(x) = \frac{1}{2}(x-3)^3 - 2 \), the \( -3 \) causes a shift of 3 units to the right.
Vertical Shifts:
Vertical shifts occur when we add or subtract a constant outside the function, moving the graph up or down. In our function, the \( -2 \) shifts the graph 2 units downwards.
Remember, the easiest way to apply these shifts is to understand them visually through the movement of each point on the graph, keeping an eye on where the graph’s key features, like vertices and inflection points, are heading.