Problem 66 - College Algebra with Corequisite Support

Question

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\frac{1}{x}\) is vertically stretched by a factor of 8 , then shifted to the right 4 units and up 2 units.

Step-by-Step Solution

Answer

The function is \( g(x) = \frac{8}{x-4} + 2 \).

Step 1: Identify the Base Function

The given toolkit function is \( f(x) = \frac{1}{x} \). We will apply the transformations described to modify this function.

Step 2: Apply Vertical Stretch

Apply the vertical stretch by a factor of 8 to the base function. This means we multiply the entire function by 8. Thus, it changes from \( f(x) = \frac{1}{x} \) to \( g(x) = 8 \cdot \frac{1}{x} = \frac{8}{x} \).

Step 3: Apply Horizontal Shift

The function is shifted right by 4 units. To accomplish this, replace \( x \) with \( (x-4) \). The formula becomes \( g(x) = \frac{8}{x-4} \).

Step 4: Apply Vertical Shift

The final step is to shift the function up by 2 units. This is done by adding 2 to the entire function. Therefore, the formula is modified to \( g(x) = \frac{8}{x-4} + 2 \).

Key Concepts

Vertical Stretch

A vertical stretch occurs when a function is expanded or stretched along the y-axis. In simpler terms, a vertical stretch takes the graph of a given function and pulls it upward, increasing the distance of all points from the x-axis. The degree of stretching is determined by a specific factor, often represented by a constant greater than 1.

Here's what you need to keep in mind about vertical stretches:

• When you apply a vertical stretch by a factor of 8, it means every y-value of the original function is multiplied by 8.
• This results in the function becoming taller or more stretched vertically.
• In mathematical notation, if you have the function \( f(x) = \frac{1}{x} \), a vertical stretch by factor 8 transforms it into \( g(x) = 8 \cdot f(x) = \frac{8}{x} \).

Always remember that stretching vertically changes the steepness of the graph, but it doesn't alter the x-values (horizontal position) of points on the graph.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function to the left or to the right along the x-axis without changing its shape. This is done by adding or subtracting from the x-value in the function's formula.

Important points about horizontal shifts:

• To shift a function to the right, you subtract from the x-variable in the function. For example, shifting \( f(x) \) right by 4 units means replacing \( x \) with \( x - 4 \).
• Conversely, to shift left, you add to the x-variable, e.g., replace \( x \) with \( x + a \), where \( a \) is the units shifted.
• In our problem, with \( g(x) = \frac{8}{x} \), shifting right by 4 units changes it to \( g(x) = \frac{8}{x-4} \).

This manipulation has the effect of moving every point on the graph to a new x-position without altering the vertical positioning. It's a straightforward way to adjust where the graph is placed on the horizontal axis.

Vertical Shift

In a vertical shift, the graph of a function is moved up or down along the y-axis. This transformation keeps the x-values unchanged while modifying the y-values by a constant amount.

Here's what you should know about vertical shifts:

• To shift a function upwards, you add a positive constant to the function. For our given transform, adding 2 moves the graph up by 2 units.
• To shift downwards, you'd subtract a constant from the function.
• So, for the function \( g(x) = \frac{8}{x-4} \), adding 2 results in \( g(x) = \frac{8}{x-4} + 2 \).

This kind of transformation is synonymous with altering the y-values of every point on the graph by the same constant, essentially lifting or lowering the entire graph without modifying its shape.