Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{3}\) to sketch the graph of the function. $$f(x)=x^{3}-2$$

Step-by-Step Solution

Answer

The graph of \(f(x) = x^{3}-2\) is just like that of the base function \(y=x^{3}\), but it's shifted down by 2 units.

Step 1: Identify the Base Function

The base function is given as \(y=x^{3}\). It's a cubic function and its graph is a curve that first decreases (for negative x), then increases. The origin is a point of inflection.

Step 2: Understand the Transformation

Subtracting 2 from the function \(x^{3}\) will result in a shift of its graph. The constant 2 subtracted from \(x^{3}\) means the function will be shifted vertically downwards by 2 units.

Step 3: Sketch the Graph

First, draw the original function's graph. The function's graph \(y=x^{3}\) goes through the origin, so mark this point. Then draw a smooth curve which first goes downwards for negative x values, and upwards for positive x values. To sketch the graph of \(f(x) = x^{3}-2\), shift the entire graph of \(x^{3}\) down by 2 units. This means every point in the original graph is moved down by 2 units in the new graph.

Key Concepts

Graph Transformations

Graph transformations are changes made to the appearance of a graph while maintaining the original function's shape. These transformations can include translations, stretches, compressions, and reflections.
In the context of cubic functions like the base function \( y=x^3 \), transformations allow us to move or alter the graph in a controlled manner. This can be helpful for understanding how changes in the equation affect the graph's shape and position.
Common types of transformations include:

• Translations: Shifts of the graph either horizontally or vertically.
• Stretches/Compressions: Alterations to make the graph narrower or wider.
• Reflections: Inversions over an axis.

For instance, in our exercise, the function \( f(x) = x^3 - 2 \) is a translation. Specifically, it involves a vertical shift, which we will discuss more in-depth in the next section.

Vertical Shift

A vertical shift is a type of graph transformation where all the points of a function's graph are moved up or down along the y-axis. This occurs when a constant is added or subtracted from the function. It doesn’t change the shape of the graph, just its position vertically.
For cubic functions like \( y = x^3 \), a vertical shift can be easily visualized. In our case, subtracting 2 in the function \( f(x) = x^3 - 2 \) indicates a shift 2 units downward. This means each point on the original graph of \( y = x^3 \) moves down by 2 units.
Understanding vertical shifts can make sketching graphs easier, as you simply take the base graph and adjust its position vertically:

• Add a positive constant to move it up.
• Subtract a constant to move it down.

This kind of transformation maintains the graph's symmetry and inflection points while changing its vertical position.

Point of Inflection

A point of inflection is a critical concept when studying graph shapes. This point is where the graph changes its concavity, which means it goes from being "concave up" to "concave down," or vice versa.
For the cubic function \( y = x^3 \), the point of inflection is at the origin (0,0). Here, the curvature switches, giving the cubic graph its distinctive S-shape.
In transformations such as vertical shifts, the point of inflection shifts with the graph. So, for \( f(x) = x^3 - 2 \), this means the point of inflection moves down to (0, -2), maintaining the graph's characteristic behavior despite its new position.
Identifying and understanding points of inflection can help predict how transformations like shifts or stretches affect the entirety of the graph. This knowledge is central to sketching and analyzing function behaviors across different transformations.