Problem 46 - A Graphical Approach to College Algebra

Question

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(x-6)$$

Step-by-Step Solution

Answer

The graph of \( y = \log_2(x-6) \) is the graph of \( y = \log_2 x \) shifted 6 units to the right.

Step 1: Identify the Parent Function

The given function is \( y = \log_2(x-6) \). The parent function is \( y = \log_2 x \), which is a logarithmic function with base 2.

Step 2: Determine the Transformation

The function \( y = \log_2(x-6) \) is a horizontal shift of the parent function \( y = \log_2 x \). The presence of \((x-6)\) indicates a shift to the right by 6 units.

Step 3: Graph the Parent Function

Begin by graphing the parent function \( y = \log_2 x \). It passes through the point (1,0), and as \( x \) approaches infinity, \( y \) increases. The graph is undefined for \( x \leq 0 \). The vertical asymptote is at \( x = 0 \).

Step 4: Apply the Horizontal Shift

Shift the entire graph of \( y = \log_2 x \) to the right by 6 units. This moves the vertical asymptote to \( x = 6 \), and the point (1,0) to (7,0).

Step 5: Sketch the Transformed Graph

Draw the transformed graph starting from the new vertical asymptote at \( x = 6 \). The graph now passes through (7,0) and continues to increase as \( x \) increases. Ensure the shape of the log curve is maintained.

Key Concepts

Logarithmic Functions

Logarithmic functions are an essential tool in mathematics, often used to model exponential growth and decay. They are inverses of exponential functions. The general form of a logarithmic function is \( y = \log_{b} x \), where \( b \) is the base of the logarithm, and \( x \) is the argument. In the context of our exercise, the parent logarithmic function is \( y = \log_{2} x \). This particular function is increasing when \( x > 0 \) and features a vertical asymptote at \( x = 0 \). That means the graph approaches a vertical line but never actually touches or crosses it.Key features of logarithmic functions include:

• They pass through the point (1,0) for \( \log_{b} x \) where \( b = 2 \), because \( \log_{b} 1 = 0 \).
• The domain is \( (0, \infty) \), while the range is \( (-\infty, \infty) \).
• The curve gradually rises to the right, getting less steep as \( x \) increases.

Understanding these properties helps in graphing and transforming logarithmic functions, which is crucial for solving exercises involving shifts and stretches.

Horizontal Shifts

Horizontal shifts refer to moving a graph left or right along the x-axis. In our exercise, we are dealing with a horizontal shift caused by the function \( y = \log_2(x-6) \). Here, the \(-6\) inside the parentheses is key.To determine the direction of the shift:

• If the function is \( y = \log_2(x-c) \), the graph shifts \( c \) units to the right.
• If the function is \( y = \log_2(x+c) \), the graph shifts \( c \) units to the left.

For our function, \( y = \log_2(x-6) \), the "-6" means we move the parent function \( y = \log_2 x \) to the right by 6 units. This transforms:

• The vertical asymptote from \( x = 0 \) to \( x = 6 \).
• The point \( (1,0) \) from the parent graph to \( (7,0) \).

Understanding horizontal shifts allows us to accurately graph functions by predicting their transformation in a straightforward manner.

Parent Functions

Parent functions are the simplest form of a function within a family of functions. They serve as the building blocks for graph transformations, such as shifts, stretches, and reflections. In the context of our exercise, the parent function is \( y = \log_2 x \).Parent functions provide a reference point:

• They help us identify transformation types.
• They enable us to predict graph behavior and transformations.
• By starting with the parent function, we can apply transformations to achieve the desired graph.

For the logarithmic family, the parent function \( y = \log_2 x \) has a vertical asymptote at \( x = 0 \) and passes through the origin point (1,0). Graphically, this creates a continuous curve that increases as \( x \) moves towards infinity.By understanding the parent function, you can intuitively apply transformations like those discussed in the horizontal shifts section. This helps in graphing more complex functions with ease.