Problem 75 - Precalculus : Building Concepts and Connections

Question

Use a graphing utility to solve the problem. Graph \(f(x)=x^{3}\) and \(g(x)=(x-7)^{3} .\) How can the graph of \(g\) be described in terms of the graph of \(f ?\)

Step-by-Step Solution

Answer

The graph of \(g(x) = (x - 7)^3\) is the same as the graph of \(f(x) = x^3\) but shifted 7 units to the right.

Step 1: Graph the Functions

Use a graphing utility to plot the two functions \(f(x) = x^3\) and \(g(x) = (x-7)^3\). Notice the shape and orientation of both graphs.

Step 2: Observe the Kinship

Observe that the graph of \(g\) is exactly the same shape as the graph of \(f\), but it is shifted to the right by 7 units. This shift occurs because we subtract 7 from \(x\) in the function \(g(x)\).

Step 3: Describe the Relationship

Describe the graph of \(g\) in relation to \(f\). The graph of \(g(x)=(x-7)^{3}\) is the graph of \(f(x)=x^{3}\) shifted 7 units to the right.

Key Concepts

Transformation of Functions

When learning about functions in algebra, one of the foundational concepts is that of transformation of functions. A function transformation takes a basic function and modifies it, resulting in a shift, stretch, compression, or reflection of its graph. Understanding transformations allow us to translate complex equations into visuals, thereby providing insight into the behavior of the function across its domain.

One of the simplest forms of transformations is a vertical or horizontal shift, which occurs when we add or subtract a constant from the function's variable. For instance, adding a constant to the function's output will shift the graph vertically, while modifying the function's input variable—as we do with a horizontal shift—will move the graph left or right. This concept is pivotal in exploring and comparing the relationships between different functions.

Horizontal Shift

A horizontal shift is a type of transformation that slides a graph left or right on the coordinate plane. When we subtract a constant from the variable in the function expression, such as going from \(f(x) = x^3\) to \(g(x) = (x - 7)^3\), we are applying a horizontal shift.

Here is how it works: the minus sign in \((x - 7)^3\) indicates a shift to the right by 7 units. Conversely, if we had \((x + 7)^3\), the graph of this function would move left by 7 units. It's important to recognize that the numerical value inside the parentheses with the variable x determines the magnitude and direction of the shift, without altering the shape of the graph.

Graphing Utility

A graphing utility is a powerful tool used in mathematics to visualize functions. Whether it's a graphing calculator, a computer program, or an online application, graphing utilities provide a practical and efficient means for plotting functions and examining their features. They are not only valuable for confirming analytic solutions but also for exploring function behaviors and transformations.

By inputting a function's equation, students can instantly see the curve or line it creates. Generally, graphing utilities come equipped with capabilities such as zooming, tracing points, and plotting multiple graphs for comparison. These features make them indispensable for exercises that involve observing changes, like how the graph of one function relates to another after undergoing transformations.

Function Relationship

Understanding the function relationship is about recognizing how one function is derived or connected to another. In our exercise, we explored the relationship between \(f(x) = x^3\) and \(g(x) = (x - 7)^3\). Through transformation, particularly the horizontal shift, we compared two cubic functions and noted that \(g\) is merely a shifted version of \(f\).Knowing how two functions relate can simplify complex problems, as it allows us to apply known characteristics from one function to another. This relationship reveals that the two functions have similar properties, such as interval of increase or decrease, symmetry, and end behavior. By linking functions in this way, students can gain deeper insights and solve problems more efficiently.