Problem 42 - A Graphical Approach to Precalculus with Limits

Question

In each of the following. (a) explain how the graph of the given finction can be obtained from the graph of \(g(x)=\log _{2} x,\) and (b) graph the function. $$f(x)=-4 \log _{2} x-8$$

Step-by-Step Solution

Answer

Stretch, reflect over x-axis, and shift down 8 units the graph of \(g(x)=\log_2 x\).

Step 1: Analyze the Base Function

Firstly, consider the given function as related to the base function \( g(x) = \log_2 x \). This is a logarithmic function with base 2, which is defined for \( x > 0 \), and it passes through point (1,0).

Step 2: Identify Vertical Shift

The term \(-8\) in the function \( f(x) = -4 \log_2 x - 8 \) indicates a vertical shift. Specifically, it means the graph will be shifted downward by 8 units.

Step 3: Determine the Vertical Stretch and Reflection

The coefficient \(-4\) in front of the logarithm indicates two transformations: a vertical stretch by a factor of 4 and a reflection over the x-axis. The graph of \( g(x) = \log_2 x \) is stretched and flipped, so its steepness changes and it is inverted.

Step 4: Graph the Transformed Function

To graph \( f(x) = -4 \log_2 x - 8 \), start by plotting key points from \( g(x) \) and apply the transformations: stretch and reflect the graph, and then shift it down 8 units. Important points like (2, -4) after transformations can help in sketching the graph.

Key Concepts

Vertical Shifts

Vertical shifts involve moving a graph up or down without changing its shape. Consider the function \(f(x) = -4 \log_2 x - 8\). The -8 part at the end of the function tells us that the graph of our function is being shifted vertically. But which way?

• A positive number results in an upward shift by that number of units.
• A negative number, like in our function, results in a downward shift by that exact amount.

For \(f(x)\), the graph will move down by 8 units. Imagine you have the basic graph of \(g(x) = \log_2x\). Every point on this graph, such as the point (1, 0), will now be moved to (1, -8). This shift maintains the same shape of the graph but relocates its position on the coordinate grid.

Vertical Stretch

A vertical stretch affects the steepness of a graph, making it either taller or flatter without moving its base position. In our function \(f(x) = -4 \log_2 x - 8\), the -4 in front of the \(\log\) signifies the vertical stretch.

• The absolute value of the number tells us the stretch factor.
• If the number is greater than 1, it indicates stretching the graph to be taller, meaning every vertical distance is multiplied by this factor.

The base function \(g(x) = \log_2 x\) becomes taller by a factor of 4. If originally the graph passed through the point (2, 1), it now passes through (2, 4), assuming no reflection or shifts. This makes our curve steeper and changes how quickly it rises or falls.

Reflection over the x-axis

Reflection over the x-axis involves flipping the graph upside down. When the word "reflection" is used in mathematics, think of a mirror placed along the x-axis. Every point above the x-axis moves an equal distance below it, and vice versa.In \(f(x) = -4 \log_2 x - 8\), there's a negative sign before the coefficient, the -4 that indicates reflection. This is what flips the graph over the x-axis.

• Reflection changes the direction of the graph.
• For instance, if a part of the graph originally rises as it moves right, it will now fall.

Using the base of \(g(x) = \log_2x\), if the point (2,1) exists on the original graph, it becomes (2, -1) after reflection. Combining this with vertical stretch and shift results in significant transformation of our original function's graph.