Problem 40 - 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. $$f(x)=\log _{2}(x-6)$$

Step-by-Step Solution

Answer

Shift the graph of \( g(x) = \log_{2} x \) 6 units right to graph \( f(x) = \log_{2}(x-6) \).

Step 1: Identify the Base Function

The function we are given is written in terms of the base function \( g(x) = \log_{2} x \). This means our base function is a logarithmic function with base 2.

Step 2: Determine the Transformation

To transform the graph from \( g(x) = \log_{2} x \) to \( f(x) = \log_{2}(x-6) \), note that the \( x \) inside the logarithm is replaced by \( x - 6 \). This indicates a horizontal shift. Specifically, it is a shift 6 units to the right.

Step 3: Graph the Base Function

Start by sketching the graph of the base function \( g(x) = \log_{2} x \). This is a typical logarithmic curve that passes through the point (1, 0) and approaches the \( y \)-axis asymptotically.

Step 4: Apply the Transformation

Shift every point on the graph of \( g(x) = \log_{2} x \) 6 units to the right to graph \( f(x) = \log_{2}(x-6) \). The vertical asymptote originally at \( x = 0 \) will now be at \( x = 6 \). The point (1, 0) on \( g(x) \) will move to (7, 0) on \( f(x) \).

Step 5: Sketch the Transformed Graph

Draw the graph of the transformed function \( f(x) = \log_{2}(x-6) \) based on the movements explained. Ensure the graph approaches the vertical line \( x = 6 \) (new asymptote) and passes through the point (7, 0). The function will be defined for \( x > 6 \).

Key Concepts

Horizontal Shift in Logarithmic Functions

When working with logarithmic functions, understanding horizontal shifts is crucial. This transformation modifies the graph's horizontal position on the coordinate plane.
The function given by \( f(x) = \log_{2}(x-6) \) differs from its base function \( g(x) = \log_{2}x \) because of a horizontal shift.
A horizontal shift happens when you have \( x - h \) inside the logarithm, where \( h \) is some constant.

• If \( h \) is positive, the graph shifts right.
• If \( h \) is negative, the graph shifts left.

For \( f(x) = \log_{2}(x-6) \), the \( x \) is replaced by \( x-6 \). This means every point on the graph of \( g(x) \) moves 6 units to the right, resulting in a shift of the entire graph along the x-axis.
Hence, the original graph is translated horizontally without any distortion in shape.

Understanding the Base Function

The base function in this context is \( g(x) = \log_{2}x \). This is a logarithmic function, where 2 is the base of the logarithm.
Logarithmic functions are nature's way of compressing large numbers into smaller, more manageable figures. Here's what you need to know about \( g(x) = \log_{2}x \):

• The graph typically passes through the point \((1,0)\) since \( \log_{2}1 = 0 \).
• It increases slowly and never touches the y-axis, creating an asymptote at \( x = 0 \).
• The function's domain is \( x > 0 \), as taking the logarithm of zero or negative numbers is undefined.

The base function is the foundation from which transformations like shifting or stretching occur. By first understanding the base function's behavior, predicting the effects of transformations becomes straightforward.

Exploring Vertical Asymptotes

Vertical asymptotes are lines where a function gets infinitely close but never actually touches. For logarithmic functions, a vertical asymptote appears where the argument of the log function turns zero.
With \( g(x) = \log_{2}x \), the vertical asymptote is naturally at \( x = 0 \), because \( \log_{2}0 \) is undefined. However, when we apply a horizontal shift as seen in \( f(x) = \log_{2}(x-6) \), the location of this asymptote changes.
Substituting \( x - 6 \) into the function causes the asymptote to move from \( x = 0 \) to \( x = 6 \). This happens because the argument of the logarithm, \( x - 6 \), becomes zero when \( x = 6 \).

• Vertical asymptotes help define the boundaries of a function's domain. For \( f(x) \), \( x \) must be greater than 6.
• This means the function won't exist on the left-hand side of this asymptote (for \( x \leq 6 \)).

Understanding vertical asymptotes ensures accurate graphing and comprehension of a function's behavior, especially in mathematical modeling and analysis.