Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of \(g\) and \(h\) relative to the graph of \(f\).$$\begin{aligned}&f(x)=x^{3}-3 x^{2}+2\\\ &g(x)=-f(x)\\\&h(x)=f(2 x)\end{aligned}$$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a reflection of the graph of \(f(x)\) over the x-axis, creating an upside-down version of \(f(x)\). The graph of \(h(x)\) appears to be squished towards the y-axis, due to a horizontal compression of \(f(x)\).

Step 1: Graphing the Given Functions

Use the graphing utility to plot the functions \(f\), \(g\), and \(h\) on the same graph. For \(f(x)=x^{3}-3 x^{2}+2\), enter directly the expression. Then for \(g(x)=-f(x)\), you must input the original function \(f(x)\) but scale the function in the y-direction by multiplying by -1. Lastly, for \(h(x)=f(2x)\), the independent variable \(x\) in the original equation \(f(x)\) should be replaced by \(2x\). This operation stretches the function in the x-direction by a factor of 0.5 or squishes by a factor of 2 depending on perspective.

Step 2: Describing the Graph of \(g(x)\)

The function \(g(x) = -f(x)\) is a reflection of the function \(f(x)\) over the x-axis. All the y-values of the function \(f(x)\) are inverted. The graph of \(g(x)\) is an upside-down version of the graph of \(f(x)\).

Step 3: Describing the Graph of \(h(x)\)

The function \(h(x) = f(2x)\) is a horizontal compression of \(f(x)\). Every \(x\)-value on \(f(x)\) is replaced by \(x/2\) on \(h(x)\). Therefore, the graph of \(h(x)\) appears to be squished towards the y-axis, compared to the graph of \(f(x)\).

Key Concepts

Reflection of functions

Reflection of functions is a concept used to understand how changing a function affects its graph visually. When you reflect a function, it flips over a specific axis. This means that each point on the original graph is mirrored across that axis.

• For example, when reflecting a function over the x-axis, like in the case of the function \(g(x) = -f(x)\), it inverts all the y-values of the function \(f(x)\).
• This results in a graph that looks like an upside-down version of the original graph.

In simpler terms, imagine placing a mirror along the x-axis and looking at the image of the graph. That's how the reflection would appear. This reversal in direction is particularly useful, as it helps in understanding symmetries and oppositions in the graph's pattern.

Horizontal compression

Horizontal compression is a transformation that affects the width of a graph. It involves changing the scale of the graph along the x-axis.

• For the function \(h(x) = f(2x)\), every x-coordinate of the original function \(f(x)\) is effectively halved.
• This action compresses the graph towards the y-axis, making it narrower.

Mathematically, this transformation is achieved by replacing \(x\) with \(2x\), or a similar operation that changes how far apart points are on the x-axis. Due to this relationship, the function appears to squeeze, which can also be described as a type of scaling. This can be useful in fitting graphs within certain boundaries or seeing how changes in variables affect outcomes on a graph.

Graphing utility

A graphing utility is a helpful tool, especially when visualizing complex mathematical functions. It allows you to efficiently plot functions and transformations, compare different functions on the same graph, and observe their effects.

• Using a graphing utility, you can easily visualize multiple functions in the same window, such as \(f\), \(g\), and \(h\).
• This practice aids in understanding how each transformation—like reflection or compression—alters the original graph.

Most graphing utilities have features that enable you to scale graphs, view them in different dimensions, and even animate changes. They provide an interactive experience for learning about graphs and transformations, offering both a way to check your work and a means to explore visual comprehension of mathematical concepts.