Problem 56 - 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 \(f\). \(g(x)=6 f(x)\)

Step-by-Step Solution

Answer

The graph of \( g(x) = 6f(x) \) is a vertical stretch of the graph of \( f(x) \) by a factor of 6.

Step 1: Identify the Original Function

The original function given is \( f(x) \). This is the base graph before any transformations are applied.

Step 2: Determine the Type of Transformation

In the function \( g(x) = 6f(x) \), the transformation involves multiplying the original function \( f(x) \) by a constant factor of 6. This type of transformation is known as a vertical stretch.

Step 3: Analyze the Vertical Stretch

A vertical stretch means that every point on the graph of \( f(x) \) is moved further away from the x-axis, making the graph taller by a factor of 6. This increases the y-values of the graph by a factor of 6, but the x-values remain unchanged.

Step 4: Describe the Resulting Graph

The graph of \( g(x) = 6f(x) \) will look like the graph of \( f(x) \), but it will be vertically stretched. Each output value \( y = f(x) \) is now \( y = 6f(x) \), which implies that if the original graph had points at \((x, y)\), the transformed graph \( g(x) \) will have points at \((x, 6y)\).

Key Concepts

Vertical Stretch

When a function undergoes a vertical stretch, it means that the entire graph of the function is stretched or stretched outward away from the x-axis. The term "vertical stretch" comes into play when each point on the graph is moved away from the x-axis by a certain factor, making the graph taller.

For example, for the function transformation given by \( g(x) = 6f(x) \), the vertical stretch occurs due to the multiplication of the original function \( f(x) \) by 6. This means that every y-value of the function is multiplied by 6, lifting the graph up proportionally.

• The graph appears stretched or elongated along the vertical axis.
• The larger the factor, the more stretched or taller the graph looks.
• A y-value that was \( y \) in the original graph of \( f(x) \), becomes \( 6y \) in the transformed graph \( g(x) \).

Understanding this concept helps in visualizing how the shape of the graph changes as such transformations are applied.

Graph of a Function

The graph of a function is a visual representation of the set of all possible outputs \( y \) for every input \( x \). Seeing the graph provides a picture of how the function behaves, how the output values change in response to different inputs, and the overall "shape" of the function.

Each function is uniquely represented by its graph, which clearly illustrates patterns such as linearity, curvature, or other trends. For instance:

• The graph could show a linear function as a straight line.
• A quadratic function would appear as a parabola.
• Exponential functions tend to have gracing curves that rise or fall sharply.

When a graph undergoes transformations, like the vertical stretch discussed earlier, these patterns remain the same, but the graph is modified. The transformation makes it crucial to comprehend how basic shifts and changes can affect the graph's appearance.

Multiplying by a Constant Factor

One of the fundamental transformations in mathematics is multiplying a function by a constant factor, which directly impacts the graph of the function. This operation modifies the function by increasing or decreasing its output values consistently across the entire graph.

When the function \( f(x) \) is multiplied by a constant factor, like in \( g(x) = 6f(x) \), several things happen:

• The output value \( y \) of the function changes to \( 6y \), essentially scaling every point on the graph vertically.
• This multiplication affects the "vertical stretch," making the function appear taller or shorter based on the factor applied.
• Importantly, the x-values remain the same, ensuring the width of the graph isn't altered.

This makes multiplying by a constant an essential concept, as it simplifies predicting and understanding how changing one part of a function affects its overall graph.