Problem 96 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the function \(g(x) = x^3 - 2\) is a downward shift of the graph of the standard cubic function by two units.

Step 1: Graph the standard cubic function

Plot the function \(f(x) = x^3\). This is known as the standard cubic function. It is a smooth curve that increases from negative infinity, goes through the origin (0,0), and increases to positive infinity.

Step 2: Understand the transformation

Examine the function \(g(x) = x^3 - 2\), and identify the transformations from the standard cubic function. The '-2' transforming the function means that the entire graph of \(f(x) = x^3\) will be shifted down by 2 units.

Step 3: Graph the transformed function

Draw the graph of the function \(g(x) = x^3 - 2\) by taking the graph of \(f(x) = x^3\) and shifting it downwards by two units. The curve will now cross the y-axis at \(y = -2\) rather than at the origin.

Key Concepts

Cubic Function

A cubic function is a polynomial function where the highest exponent of the variable is 3. This means the general form of a cubic function is written as \(f(x) = ax^3 + bx^2 + cx + d\). In this formula:

• The term \(ax^3\) is what makes it a cubic function.
• The coefficients \(a, b, c,\) and \(d\) are constants that determine the specific shape and position of the graph.
• If \(a > 0\), the graph rises to the right, whereas if \(a < 0\), it falls to the right.

The most basic cubic function is \(f(x) = x^3\). This creates an S-shaped curve that passes through the origin (0,0) and extends infinitely in both the positive and negative directions. The graph is symmetrical about the origin and demonstrates points of inflection, where the curve changes its direction of bending.

Graph Transformations

Graph transformations allow us to change the position and orientation of a function's graph. In the case of cubic functions like \(f(x) = x^3\), various types of transformations can be applied:

• Translation: Shifting the graph horizontally or vertically.
• Reflection: Flipping the graph over an axis.
• Scaling: Stretching or compressing the graph vertically or horizontally.

In our context, the graph of \(f(x) = x^3\) is being vertically shifted, which is a type of translation. This involves moving the entire graph up or down without altering its shape or orientation. Graph transformations are an efficient way to derive new functions from existing ones by applying certain algebraic modifications.

Vertical Shift

A vertical shift occurs when every point on a graph moves the same distance up or down. In the cubic function \(g(x) = x^3 - 2\), the transformation is a vertical shift of 2 units downward compared to the standard cubic function.Here's how to spot a vertical shift in a polynomial function:

• Look for a constant term added or subtracted from the function.
• If the constant is subtracted, the graph shifts downward. If added, the graph shifts upward.
• The point where the graph crosses the y-axis (the y-intercept) will change due to this shift.

For \(g(x) = x^3 - 2\), this means the y-intercept moves from 0 at the origin to -2 on the y-axis. The entire graph maintains its overall shape, moving in a uniform downward direction.

Polynomial Functions

Polynomial functions encompass a broad category of algebraic expressions that consist of variables raised to a power and associated coefficients. The general expression for a polynomial is \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Here:

• The highest power of \(x\) determines the degree of the polynomial.
• Cubic functions are third-degree polynomials, meaning the highest exponent is 3.
• The coefficients \(a_n\) influence the graph's shape and direction.

Polynomial functions can be simple or complex, depending on the degree and coefficients. They are foundational in algebra, with graph behaviors dictated by their degree. Higher-degree polynomials exhibit more inflections and crossings. Understanding polynomial behavior provides a keen insight into the graph's dynamics, allowing for complex function manipulations like those seen in graph transformations.