Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the transformation of the graph of \(f\) that yields the graph of \(g.\) $$f(x)=\log _{8} x, \quad g(x)=-2+\log _{8}(x-3)$$

Step-by-Step Solution

Answer

The graph of \(g(x) = -2 + \log_8{x - 3}\) can be obtained from the graph of \(f(x) = \log_8{x}\) by shifting the graph of \(f(x)\) \(3\) units to the right and then \(2\) units downward.

Step 1: Identify Vertical Shift

The graph of \(g(x)\) is obtained from the graph of \(f(x)\) by subtracting a constant to \(f(x)\). Since \(g(x) = -2 + \log_8{x - 3}\), it can be seen that \(2\) has been subtracted from \(f(x)\). This represents a vertical shift of \(2\) units downward.

Step 2: Identify Horizontal Shift

The argument of the log function in \(g(x)\) is \(x - 3\). This means that the graph of \(f(x)\) has been shifted to the right by \(3\) units to produce the graph of \(g(x)\).

Step 3: Combine Transformations

Put together both transformations. The graph of \(g(x)\) is obtained by taking the graph of \(f(x)\), shifting it \(3\) units to the right and then \(2\) units downward.

Key Concepts

Vertical Shift

When we talk about vertical shifts in graph transformations, we're describing how far up or down a graph moves on a coordinate plane. For instance, if you have a basic function, and you add or subtract a number to it, you shift the graph up or down. Let’s connect this with our example, where the original function is \( f(x) = \log_8 x \), and the transformed function is \( g(x) = -2 + \log_8 (x - 3) \).

• Here, the \( -2 \) before the logarithm indicates a downward vertical shift.
• Adding \( -2 \) moves every part of the graph \( 2 \) units lower.

It's important to recognize the sign here. A negative number moves the graph down, while a positive number would move it up.

Horizontal Shift

Horizontal shifts occur when we move a graph left or right along the x-axis. This shift happens when a function’s input variable (often x) is adjusted. In our logarithmic function example, the shift is related to the expression within the logarithm. The function \( g(x) = -2 + \log_8 (x - 3) \) derives its horizontal shift from the modification inside the log expression: \( x - 3 \).

• The \( -3 \) tells us the graph moves right by 3 units.
• If it had been \( x + 3 \), we'd move left instead.

Why does \( x - 3 \) mean move to the right? It’s all about how the input to your function changes where the graph starts. By subtracting from \( x \), it implies we need a larger x-value to reach the same y-values as before, hence moving rightward.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They help us find the power to which a base number is raised to achieve a certain value. In our example, the function\( f(x) = \log_8 x \) uses base 8. When you see \( \log_8 x \), it asks, "To what power must 8 be raised, to result in x?"

Often seen in equations representing exponential growth or decay, logarithmic functions follow specific transformation rules just like other types of functions.

• Graphing logarithmic functions usually results in curves that approach vertical asymptotes.
• Transformations apply to these functions similar to linear and quadratic functions.

By understanding how vertical and horizontal shifts alter the position of logarithmic graphs, you gain better tools to handle these and related problems in various mathematical contexts.