Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=\log _4 x, g(x)=3 \log _4 x-5\)

Short answer

In transformation of the base function \( f(x)=\log _4 x \) to the function \( g(x)=3 \log _4 x-5 \), there is a vertical stretch by a factor of 3 and a vertical shift down by 5 units.

Step-by-step solution

Identify the base functions

We start by identifying our base function \( f(x)=\log _4 x \) and transformed function \( g(x)=3 \log _4 x-5 \).

Describe the transformation

For the function \( g(x) = 3 \log _4 x - 5 \), it indicates two transformations from the base function \( f(x) = \log _4 x \). The '3' Multiplier indicates that the function will be vertically stretched by a factor of 3. The '-5' indicates that the function will be shifted down by 5 units.

Graph the functions

Using a graphing tool, plot the two functions \( f(x) = \log _4 x \) and \( g(x) = 3 \log _4 x - 5 \) on the same set of axes. Remember that logarithmic functions exhibit a vertical asymptote at x = 0, as \(\log _4 0\) is undefined. The graph of \( g(x) \) will be steeper and lower than that of \( f(x) \), due to the described transformations.

Key concepts

Vertical Stretching

Vertical stretching is a transformation that alters the appearance of a function by increasing its vertical size. This happens when we multiply the output of a function by a constant factor greater than 1. In our exercise with the function \( g(x) = 3 \log_4 x - 5 \), the multiplier 3 causes a vertical stretch.

This means every y-value of the original function \( f(x) = \log_4 x \) becomes three times as large in magnitude, making the graph appear taller or steeper. Let's break it down further:

• For every point \((x, y)\) on the graph of \(f(x)\), the corresponding point on the graph of \(g(x)\) is \((x, 3y)\).
• This stretching doesn't change the x-values, as the transformation only affects the vertical aspect.

Vertical stretching is a way to visualize how the logarithmic curve reacts when its range (outputs) are multiplied by a constant, stretching the graph vertically.

Vertical Shift

Vertical shifts occur when a constant is added to or subtracted from a function. This moves the graph up or down without altering its shape. In our example, the function \( g(x) = 3 \log_4 x - 5 \) involves a vertical shift downwards by 5 units.

Here’s how the vertical shift is implemented:

• The '-5' at the end of the function tells us to move every point on the graph of the new function down by 5 units.
• For a point at \((x, y)\), the corresponding transformed point becomes \((x, y-5)\).
• Vertical shifts do not affect horizontal position; all x-values remain unchanged.

By shifting the original logarithmic graph vertically, we can adjust its position to better match data or desired outputs while maintaining its inherent shape.

Graphing Logarithmic Functions

Understanding how to graph logarithmic functions helps us visualize their behavior and transformations like stretching and shifting. For the function \( f(x) = \log_4 x \), it's vital to first recognize that logarithmic graphs:

• Have a vertical asymptote where the argument of the logarithm is zero. In this context, it's at \(x=0\) because \(\log_4 0\) is undefined.
• Pass through the point \((1, 0)\) since any log function \(\log_b 1\) is zero.

When graphing the transformation \( g(x) = 3 \log_4 x - 5 \):

• First, apply the vertical stretch, making the graph steeper.
• Next, apply the vertical shift to adjust the entire graph downward by 5 units.

Keeping these transformations in mind, the graph becomes a visually clear representation of the changes, instantly indicating how adjustments in function transformation terms affect the overall shape and location of the graph. Drawing the graph, you now observe a stronger slope due to the vertical stretch and a lower position because of the vertical shift.