Problem 66 - College Algebra

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 Toolkit Function

The given toolkit function is the reciprocal function defined as \(f(x) = \frac{1}{x}\). This function will be subject to various transformations.

Step 2: Apply Vertical Stretch

A vertical stretch of a factor of 8 means we multiply the entire function by 8. So, the function becomes \(g(x) = 8\cdot f(x) = \frac{8}{x}\).

Step 3: Apply Horizontal Shift

A shift to the right by 4 units involves replacing \(x\) with \(x - 4\) in the function. So, the function becomes \(g(x) = \frac{8}{x-4}\).

Step 4: Apply Vertical Shift

A shift up by 2 units means adding 2 to the entire function. Thus, the function transforms to \(g(x) = \frac{8}{x-4} + 2\).

Key Concepts

Vertical Stretch

When we talk about a vertical stretch, we're discussing how a graph becomes "stretched" vertically. This occurs when we multiply the function by a constant factor greater than 1.
For instance, consider the function we've seen, \(f(x) = \frac{1}{x}\). If we vertically stretch this graph by a factor of 8, the function becomes \(g(x) = 8 \cdot f(x) = \frac{8}{x}\). What this means is that every point on the graph is now 8 times as far from the x-axis than it was initially.

• The y-values of the graph become 8 times larger.
• The graph looks steeper and taller.

Vertical stretches maintain the shape of the graph while changing its height.

Horizontal Shift

A horizontal shift changes the position of the graph along the x-axis without altering its shape. If we shift a graph right or left, we are essentially moving every point on the graph that direction.
For the function \(g(x) = \frac{8}{x}\), to shift it 4 units to the right, we adjust the formula to \(g(x) = \frac{8}{x-4}\).

• Replacing \(x\) with \(x-4\) moves the graph 4 units to the right.
• If instead, it was \(x+4\), the graph would move 4 units to the left.

This does not affect the y-values directly; it shifts the entire graph horizontally.

Vertical Shift

Vertical shifts move the graph up or down along the y-axis. This is achieved by adding or subtracting a constant value from the function.
For the function \(g(x) = \frac{8}{x-4}\), a shift up by 2 units is performed by adding 2 to the function: \(g(x) = \frac{8}{x-4} + 2\).

• Adding a positive number moves the graph upwards.
• Subtracting will move it downwards.

This alteration affects the y-values directly, effectively lifting the entire graph higher without altering its shape.

Reciprocal Function

The reciprocal function is the formula \(f(x) = \frac{1}{x}\). It's a key toolkit function with distinct characteristics.

• Its graph is a hyperbola, split into two branches.
• The x-axis and y-axis are asymptotes the graph approaches but never touches.

The reciprocal function showcases behavior where as \(x\) gets larger, \(f(x)\) gets smaller, and vice versa. Transformations such as vertical stretch, horizontal shift, and vertical shift modify the position and appearance of this graph, but its basic hyperbolic shape remains.