Problem 38 - 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. $$ g(x)=(x-2)^{3} $$

Step-by-Step Solution

Answer

The function \( g(x) = (x-2)^3 \) is a standard cubic function shifted to the right by 2 units. To graph it, one would graph the standard cubic function and then shift every point 2 units to the right.

Step 1: Graph the standard cubic function

The first step is to graph the standard cubic function, which is \( f(x) = x^3 \). This function gives a curve that passes through the origin (0, 0) and increases to the right and to the left as \(x\) becomes more positive or negative respectively.

Step 2: Understand the transformation

Next, it's important to understand the transformation. The given function is \( g(x) =(x-2)^3 \). The \( (x-2) \) expression inside the function indicates a shift or translation. Since we are subtracting 2 from \(x\), the graph of the function will shift to the right by 2 units.

Step 3: Graph the transformed function

Finally, graph the transformed function, \( g(x) =(x-2)^3 \), by taking the graph of \( f(x) = x^3 \) and shifting every point two units to the right. Basically, if there was a point at (a, a^3) on the graph of \(f\), then now there is a point at (a+2, (a+2)^3) on the graph of \(g\).

Key Concepts

Function Transformation

A function transformation involves changing a function's rules to alter its graph. Transformations can include translations, stretches, compressions, and reflections. For a cubic function, these can affect how steep the curve appears or its position on the graph.

In the context of the cubic function, transformations are systematic and predictable. When you alter the function from its basic form, say from \( f(x) = x^3 \) to another transformed state such as \( g(x) = (x-2)^3 \), you're shifting its position on the graph.

This particular change in the function represents a **translation**, moving the graph horizontally without altering its shape. Transformations like this maintain the symmetry and general appearance of the cubic function, which remains a curve passing through key points.

Graphing Functions

Graphing a function means plotting its outputs on a coordinate system and drawing the curve or line that best represents these points. The process gives a visual perspective of a function's behavior and characteristics.

For a cubic function \(f(x) = x^3\), graphing starts with identifying how it naturally behaves. Its basic graph passes through the origin (0,0) and extends infinitely in both the positive and negative directions. As the variable \(x\) grows larger, the function values accelerate upward or downward rapidly.

Plotting a cubic function:

• Determine key points, such as where the graph crosses every axis.
• Observe its symmetrical nature - for \(f(x) = x^3\), the point \(-a\) is mirrored in \(a\).
• Understand its smooth, continuous increase and decrease as you move left and right from the origin.

This provides a foundational grid before applying any transformations.

Translation of Functions

Translation in functions refers to moving the graph of a function in certain directions. For instance, you can slide a graph vertically or horizontally by altering its formula.

When you see a change from \( f(x) = x^3 \) to \( g(x) = (x-2)^3 \), this is a horizontal translation. Specifically, the graph shifts **right** by 2 units. This is counterintuitive since you subtract 2 in \((x-2)\), but it reflects a rightward move.

Key aspects of horizontal translation:

• The entire graph moves uniformly; every point shifts the same distance to the right.
• Important characteristics like shape and orientation remain unchanged.
• A shift does not affect the vertical component or values of the function.

So, a point at \((a, a^3)\) on \(f\) becomes \((a+2, (a+2)^3)\) on \(g\). This clarity helps to predict how a function will appear on a graph after translation.