Problem 44 - Algebra and Trigonometry

Question

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{5} x^{3}-1 $$

Step-by-Step Solution

Answer

Start with \( y = x^3 \), compress vertically by \( \frac{1}{5} \), then shift down 1 unit.

Step 1: Recognize the Standard Function

Start by recognizing the standard function that resembles the given function. Here, the given function is \( y = \frac{1}{5}x^3 - 1 \). The standard function is \( y = x^3 \).

Step 2: Apply Vertical Stretch/Compression

The coefficient \( \frac{1}{5} \) in front of \( x^3 \) suggests a vertical compression. Normally, the graph of \( y = x^3 \) passes through (1, 1), but due to the \( \frac{1}{5} \) factor, it will pass through (1, \( \frac{1}{5} \)) instead. Thus, the graph will rise slower and descend slower than the standard \( x^3 \) graph.

Step 3: Apply Vertical Shift

The \(-1\) subtracts 1 from the function, which translates the entire graph 1 unit downward. So, every point on the function \( \frac{1}{5}x^3 \) is moved lower by 1 unit.

Step 4: Sketch the Transformed Graph

Combine the transformations to sketch the graph. Begin with a rough sketch of \( y = x^3 \), apply the vertical compression resulting in a shallower curve at the origin, and then shift the graph downward by 1 unit. The turning points and behavior at infinity remain similar to \( y = \frac{1}{5}x^3 \), just translated downward.

Key Concepts

Vertical Compression

Vertical compression is an important concept when it comes to transforming the graph of a function. This concept involves the squishing or compressing of the graph's height. In the original function \( y = \frac{1}{5}x^3 - 1 \), the coefficient \( \frac{1}{5} \) applied to \( x^3 \) implies a vertical compression.

• Vertical compression occurs when the absolute value of the multiplier (in this case, \( \frac{1}{5} \)) is between 0 and 1.
• The smaller value results in the graph appearing flatter than the original.
• This effect reduces the vertical stretch, allowing the graph to rise and fall more slowly.

In practical terms, if you normally have a point at (1,1) for \( y = x^3 \), a compression would place it at (1, \( \frac{1}{5} \)), flattening and lowering the height of the curve as you move away from the origin.

Vertical Shift

Vertical shift is the transformation that moves the entire graph up or down along the y-axis. In the function \( y = \frac{1}{5}x^3 - 1 \), the term \(-1\) indicates a downward vertical shift.

• A positive constant would move the graph up.
• A negative constant, as in our example, lowers the graph by that value.

Here, subtracting 1 from the function shifts each point of the graph one unit downward. This doesn’t affect the shape or flatness of the graph; rather, it simply translates its position. So, the entire graph of the function \( \frac{1}{5}x^3 \) shifts down by one unit, maintaining its compressed form.

Cubic Functions

Cubic functions are polynomial equations of the form \( y = ax^3 + bx^2 + cx + d \). They often have distinctive shapes characterized by their curves.

• For the basic cubic \( y = x^3 \), the graph runs from lower left to upper right.
• It has a single point of inflection at the origin where the graph changes curvature.

In transformations, the basic cubic function is the point of departure. The standard cubic \( x^3 \) serves as the template, and transformations such as vertical compression or shift modify the curve without changing this fundamental behavior. Cubic functions, including their transformations, showcase predictable end-behavior where they extend towards positive and negative infinity.

Sketching Graphs

Sketching graphs requires understanding both the original function and its transformations. When sketching \( y = \frac{1}{5}x^3 - 1 \), the process comprises several steps to ensure accuracy and clarity.

• Start with the basic cubic function \( y = x^3 \).
• Apply the vertical compression to reflect the \( \frac{1}{5} \) multiplier.
• Shift the graph down by 1 unit.

As you sketch, ensure the curve is flatter due to the vertical compression, showing that the graph does not ascend or descend as sharply as the usual cubic function. Finally, make sure the entire graph is positioned one unit lower on the y-axis. This methodical approach helps in visualizing how each transformation affects the original function, aiding in drawing an accurate representation.