Problem 97 - 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. $$g(x)=(x-3)^{3}$$

Step-by-Step Solution

Answer

The function \(g(x) = (x - 3)^3\) is a transformation of the standard cubic function \(f(x) = x^3\), specifically a horizontal shift three units to the right. Graph this function by plotting \(f(x) = x^3\) then shifting the graph three units to the right.

Step 1: Graphing the standard cubic function

Start off by graphing the standard cubic function \(f(x) = x^3\). This can be done using a graphing tool, or a plotted graph. The curve of \(f(x) = x^3\) starts from bottom left, crossing through the origin (0,0), and ends at top right. The curve is an increasing function.

Step 2: Identifying the transformation

The given function is \(g(x) = (x - 3)^3\). This function represents a horizontal shift of the standard cubic function \(f(x) = x^3\). The '3' inside the parentheses indicates the shift 3 units to the right.

Step 3: Graphing the transformed function

Finally, graph the function \(g(x) = (x - 3)^3\) by shifting the graph of \(f(x) = x^3\) three units to the right. The curve of \(g(x) = (x - 3)^3\), like \(f(x) = x^3\), starts from bottom left, however it crosses at point (3,0) due to the shift, and ends at top right. The curve is still an increasing function.

Key Concepts

Graph Transformations

When we hear about graph transformations, it often seems complex, but it’s actually just a way to move and reconfigure graphs using specific rules. A graph transformation involves manipulating the graph of a function in order to achieve the graph of another, related function. It's like taking a picture and adjusting its position or size. In mathematics, these transformations fall into several types:

• Translation, which moves the graph without rotating or resizing it.
• Reflection, which flips the graph over a specific line, like the x-axis or y-axis.
• Dilation, which changes the size of the graph either vertically or horizontally.

For the cubic function example, we specifically utilize one of the translation transformations. The graph of the original function is taken, and by applying appropriate translation, it is moved horizontally or vertically, depending on the specific transformation rule provided.

Horizontal Shift

A horizontal shift is a specific type of graph transformation where the graph moves left or right along the x-axis. If you have ever dragged an object from one side to another on a screen, then you know what a horizontal shift feels like. For a function like our cubic function, shifting the graph horizontally is the key transformation applied.For instance, the cubic function \(f(x) = x^3\) can be transformed into \(g(x) = (x - 3)^3\) by implementing a horizontal shift. The \'3\' in \((x - 3)^3\) indicates this shift. Here’s how it works:

• If the form is \((x - c)^3\), shift to the right by \(c\) units.
• If the form is \((x + c)^3\), shift to the left by \(c\) units.

This transformation does not change the shape of the graph, only its position on the plane, resulting in a graph of \(g(x)\) that starts at point (3, 0) and continues with the same increasing pattern as \(f(x)\). This is an essential concept in understanding how small adjustments in a function's equation will impact the graph itself.

Polynomial Functions

Polynomial functions are a cornerstone in algebra and calculus, made up of terms consisting of a variable raised to various powers with coefficients. A cubic function like \(f(x) = x^3\) is a type of polynomial function where the highest degree of the variable is three.Here are the key aspects of polynomial functions you should know:

• Standard Form: Polynomial functions are often presented in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\).
• Degree: The degree of a polynomial is the highest exponent on the variable. In cubic functions, this would be three.
• Behavior at Endpoints: Determines the graph's direction as \(x\) approaches infinity or negative infinity. Cubic functions tend to have distinctive S-shaped curves.

Understanding polynomial functions is crucial because they describe patterns within data, other mathematical relationships, and real-world processes smoothly and seamlessly. While cubic functions are just one type of polynomial, mastering them gives a glimpse into the broader world of polynomial behavior.