Problem 1 - Algebra 2 and Trigonometry

Question

Megan said that if \(a>1\) and \(g(x)=\frac{1}{a} \mathrm{f}(x),\) then the graph of \(\mathrm{f}(x)\) is the graph of \(\mathrm{g}(x)\) stretched vertically by the factor \(a\) . Do you agree with Megan? Explain why or why not.

Step-by-Step Solution

Answer

No, Megan is incorrect. \( g(x) = \frac{1}{a}f(x) \) compresses the graph vertically by factor \(a\).

Step 1: Understand the Functions

Megan is comparing two functions: \( f(x) \) and \( g(x) = \frac{1}{a}f(x) \). The goal is to determine how the transformation from \( f(x) \) to \( g(x) \) affects the graph.

Step 2: Consider Vertical Stretching

A vertical stretch by a factor of \(a\) implies multiplying the function's output by \(a\) to form \( a \cdot f(x) \). This transformation causes the points on the graph of \( f(x) \) to move farther away from the x-axis by \(a\) times.

Step 3: Analyze Function Transformation

In the given \( g(x) = \frac{1}{a}f(x) \), each y-coordinate of \( f(x) \) is divided by \(a\). This results in y-coordinates that are \( \frac{1}{a} \) times those of \( f(x) \), which is actually a vertical compression, not a stretch.

Step 4: Draw Conclusion

The transformation \( g(x) = \frac{1}{a}f(x) \) compresses the graph vertically by a factor of \(a\), rather than stretching it. Megan's statement is incorrect.

Key Concepts

Vertical Stretching

Vertical stretching occurs when you multiply a function by a factor greater than one. In mathematical terms, if we have a function \( f(x) \), and we multiply it by a constant \( a > 1 \), we get a new function \( a \, f(x) \). This operation stretches the original graph of \( f(x) \) away from the x-axis.
Consider a simple function like \( f(x) = x^2 \). When we apply a vertical stretch using a factor of 3, the function becomes \( 3x^2 \).

• The y-values of points on the graph are 3 times higher than the corresponding y-values of \( f(x) \).
• This makes the graph look taller and narrower, as if you've pulled it upwards, while the x-values remain unchanged.

In graph transformations, vertical stretching changes the graph in such a way that it affects the steepness, but not the basic shape.

Vertical Compression

Vertical compression is the opposite of vertical stretching: instead of pulling the graph away from the x-axis, you bring the graph closer to it. This occurs when the function \( f(x) \) is multiplied by a factor between 0 and 1. For example, if we have \( g(x) = \frac{1}{a}f(x) \) where \( a > 1 \), each y-coordinate of the function \( f(x) \) is divided by \( a \).
This operation reduces the distance each point on \( f(x) \) is from the x-axis, resulting in a squished effect.

• The graph appears wider and shorter, making it seem as if it has been compressed vertically.
• Important note: while the graph is compressed vertically, the x-values remain the same.

Vertical compression changes the steepness of the graph but retains its original shape.

Graph Transformations

Graph transformations involve changing the appearance or position of a graph without altering the equation's integrity. These transformations can be shifts, stretches, or compressions in different directions. Vertical transformations—vertical stretching and compression—adjust the y-values, thereby altering the graph's vertical appearance.

When tackling these transformations:

• A vertical stretch multiplies all output values by the factor of stretch.
• Vertical compression divides all output values by the factor.
• These transformations affect only the y-values and leave the x-coordinates unaffected.

Understanding these changes is crucial for mastering the manipulation of functions and recognizing their graphical representation.

Function Comparison

Function comparison helps to understand the effects of transformations. By comparing the original function to its transformed counterpart, you can see how the adjustments affect the graph.

In the example given, we compared \( f(x) \) to \( g(x) = \frac{1}{a}f(x) \).
Megan initially thought \( g(x) \) was a vertical stretch by a factor, but it was actually a vertical compression.

• To compare, check each point on the functions to see how they move relative to the transformations applied.
• Always verify whether the function is being multiplied or divided by the factor, which affects whether it is stretched or compressed.

Precise evaluation of these transformations aids in avoiding misconceptions and accurately determining graph changes.