Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). \(g(x)=f\left(\frac{1}{3} x\right)\)

Step-by-Step Solution

Answer

The graph of \( g(x) = f\left(\frac{1}{3}x\right) \) is a horizontal compression of \( f(x) \) by a factor of 3.

Step 1: Identify the Original Function

The problem provides the function transformation of the original function \( f(x) \). We need to interpret \( f(x) \) as our base graph that will undergo transformation. The graph of \( f(x) \) can initially be any given function, such as a parabola, line, or any continuous curve.

Step 2: Identify the Transformation

The function provided is \( g(x) = f\left(\frac{1}{3}x\right) \). This transformation involves replacing \( x \) with \( \frac{1}{3}x \). This is characterized as a horizontal transformation affecting how the graph of \( f \) is displayed horizontally.

Step 3: Determine the Type of Transformation

The transformation \( f\left(\frac{1}{3}x\right) \) signifies a horizontal scaling. When \( x \) is replaced by \( \frac{1}{k}x \) for \( k > 1 \), the function is horizontally stretched. Conversely, if \( 0 < k < 1 \), which is the case here (where \( k = 3 \)), the function undergoes a horizontal compression by a factor of 3.

Step 4: Describe the Transformation Effect on the Graph

This horizontal scaling means each \( x \)-value in the original graph \( f(x) \) is moved closer to the \( y \)-axis by a factor of 3. Essentially, for any point \((x, y)\) on the original graph, the corresponding point on the transformed graph will be \(\left(\frac{x}{3}, y\right)\). Thus, the graph appears squeezed horizontally by a factor of 3.

Key Concepts

Horizontal Scaling

Horizontal scaling is a transformative process in which the function's graph's width is affected. When we discuss scaling horizontally, we are examining how changes to the input variable, \( x \), impact the function's appearance. If we have a function \( f(x) \) and we replace \( x \) with \( \frac{1}{k}x \), the graph will undergo a horizontal scaling. This can either be a stretch or a compression, depending on the value of \( k \).

• For \( k > 1 \), the function experiences a horizontal stretch. In other words, it widens. Each point on the graph seems to be pushed further from the \( y \)-axis.
• For \( 0 < k < 1 \), the function undergoes a horizontal compression, meaning it becomes narrower. Points on the graph move closer to the \( y \)-axis.

It's important to visualize how the graph changes: imagine stretching or compressing a rubber band either to the sides or closer at the center. This visual can help in understanding the effect more thoroughly.

Graphing Functions

Graphing functions involves plotting the output values \( y \) that correspond to the input values \( x \) on a coordinate system. The essence of graphing is to visually depict how one variable depends on another. Here are some key points to remember:

• The original function, \( f(x) \), is our starting point. It could be any type of function, such as linear, quadratic, or exponential.
• Transformations like horizontal scaling affect how this graph looks but do not change the overall shape of the function, just its spread across the \( x \)-axis.
• Always start by determining critical points such as intersections with axes or vertices on the graph.
• Observe how transformations modify these critical points as you adjust \( x \) values for transformation.

Graphing transformed functions helps identify changes in the graph's orientation or dimensions, aiding in better understanding of the function's behavior.

Horizontal Compression

Horizontal compression of a function graph, like the one in the exercise, occurs when you have \( g(x) = f\left(\frac{1}{3}x\right) \). Here, the value \( \frac{1}{3} \) compresses the graph toward the \( y \)-axis by a factor of 3. This means every point along the \( x \)-axis is three times closer than in the original function \( f(x) \).To understand better:

• For every point on \( f(x) \), \( (x, y) \), its new position on \( g(x) \) will be \( \left(\frac{x}{3}, y\right) \).
• Horizontal compression suggests that the function's width along the \( x \)-axis is reduced.

Imagine taking the original graph and pushing the sides inwards toward the center. This compresses all points closer, altering how the function spreads across the graph. In applying these transformations, always pay attention to how they affect the location of key points on the graph to ensure accurate representation.