Problem 36 - 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. $$ g(x)=x^{3}-2 $$

Step-by-Step Solution

Answer

The function \(g(x)=x^{3}-2\) is a graph of the standard cubic function \(f(x)=x^{3}\) shifted downwards by 2 units. This is done by subtracting 2 from each of the \(f(x)\) values.

Step 1: Plotting the Standard Cubic Function

Start by plotting the standard cubic function \(f(x)=x^{3}\). Assign different values to \(x\) and calculate the corresponding \(f(x)\) values. The points \((-2,-8)\), \((-1,-1)\), \((0,0)\), \((1,1)\), and \((2,8)\) can be included in this graph.

Step 2: Understanding the Transformation

Next, identify the transformation that is performed on the base function. The function \(g(x)=x^{3}-2\) is obtained by shifting the graph of the function \(f(x)=x^{3}\) downwards by 2 units.

Step 3: Plotting the Transformed Function

Now plot the function \(g(x)=x^{3}-2\). A quick way to do this is to shift every point on the graph of \(f(x)\) down by two units. The points would then become \((-2,-10)\), \((-1,-3)\), \((0,-2)\), \((1,-1)\), and \((2,6)\). This is the graph of \(g(x)=x^{3}-2\).

Key Concepts

Cubic Function

The cubic function is an important type of polynomial function modeled as \(f(x) = x^3\). It is characterized by its distinct shape, often referred to as a 'S' curve. Cubic functions have three main parts: rights, sharp turns, and increasing steepness as they move away from the x-axis. Unlike quadratic functions, which form a parabolic shape, cubic functions have a more extended reach along the y-axis.

• They pass through the origin, which is the point (0,0).
• Cubic graphs are symmetric in nature, around the origin.
• They display points of inflection - where the curve changes curvature.

Understanding cubic functions helps in deciphering more complex polynomial functions. You often start by graphing the simple cubic function before proceeding with transformations or examining their behavior.

Function Transformation

Function transformation involves shifting, reflecting, stretching, or compressing the graph of a given function. These transformations take a baseline function and alter its shape or position without changing the nature of the function. In the case of a cubic function, transformations help in moving it around the coordinate plane to represent real-world data effectively.
Transformations come in various forms:

• Vertical shifts, where the entire graph shifts up or down.
• Horizontal shifts, moving the graph left or right.
• Stretching or compressing, which alters the width or steepness of a curve.
• Reflection, flipping the graph across an axis.

By understanding these transformations, one can manipulate a function like \(f(x) = x^3\) and create other functions with similar properties but different positions on the graph.

Vertical Shift

A vertical shift is one of the simplest types of transformation, where the graph of the function moves up or down along the y-axis. This move is achieved by adding or subtracting a constant from the output of the base (original) function.
For the given function \(g(x) = x^3 - 2\), the cubic function \(f(x) = x^3\) is vertically shifted down by 2 units. This means each point on the original cubic graph is moved downward, lowering its y-coordinate by 2.

• If the constant is positive, the graph shifts up.
• If the constant is negative, the graph shifts down.

In this way, a vertical shift is an easy yet crucial tool for adjusting the positioning of graphs, ensuring they align with specific data or behaviors you wish to represent visually.