Problem 100 - College Algebra

Question

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f.\) $$f(x)=\log x, g(x)=\log (x-2)+1$$

Step-by-Step Solution

Answer

The graph of \(g(x) = \log (x-2) + 1\) is exactly the same shape as the graph of \(f(x) = \log x\), but it is shifted 2 units to the right on the x-axis and 1 unit up on the y-axis

Step 1: Sketching f(x)

The graph of the function \(f(x) = \log x\) is a simple logarithmic curve. As \(x\) tends towards 0 from the positive side, \(f(x)\) tends towards negative infinity. When \(x=1\), \(f(x) = 0\). For \(x > 1\), \(f(x)\) increases slowly.

Step 2: Sketching g(x)

The function \(g(x) = \log (x-2) + 1\) is the transformation of the function \(f(x) = \log x\). Here, from the function \(f(x)\), it shifts 2 units to the right on the x-axis, and 1 unit up on the y-axis.

Step 3: Comparing the graphs

Upon comparing the graphs, it is evident that the graph \(g(x)\) is exactly the same shape as the graph \(f(x)\), i.e. a logarithmic curve, but it is shifted 2 units to the right and 1 unit up. This comes from the changes inside the logarithm and the addition outside the logarithm in the function \(g(x)\).

Key Concepts

Logarithmic Functions

Understanding logarithmic functions can be quite simple once you get the hang of it. A logarithmic function is the inverse of an exponential function. The basic form is \(f(x) = \log_b(x)\), where \(b\) is the base of the logarithm. In most cases, such as the problem we're looking at, the base is 10 (common logarithm) or \(e\) (natural logarithm).

Logarithmic functions have some neat characteristics:

• They pass through the point \((1,0)\) because \(\log_b(1) = 0\).
• The domain is \(x > 0\) and the range is all real numbers.
• The curve approaches negative infinity as \(x\) approaches 0 from the right.
• As \(x\) increases, the curve increases, but at a decreasing rate.

In our exercise, \(f(x) = \log x\) perfectly illustrates these traits. Notice how it starts steep near zero and flattens as \(x\) grows. This gives the famous curve of the logarithm.

Function Shifts

Function shifts are a straightforward way to manipulate a graph's position without altering its basic shape. They can make visualizing more complex functions far easier by translating the graph along the axes.

We have two main types of shifts:

• Horizontal shifts: When you add or subtract a constant directly inside the function (e.g., \(g(x) = \log(x-2)\)), the graph moves left or right. In this case, \(g(x)\) shifts 2 units to the right compared to \(f(x)\).
• Vertical shifts: When you add or subtract a constant outside of the function (e.g., \(g(x) = \log(x-2) + 1\)), it shifts up or down. Here, \(g(x)\) has been shifted 1 unit upwards.

These shifts mean that although \(g(x)\) is based on \(f(x)\), it's located at a different spot on the coordinate plane, specifically 2 units to the right and 1 unit up.

Graphing Techniques

Graphing techniques are key tools in visualizing and understanding mathematical functions. For logarithmic functions, some strategies help make sense of their transformations and shifts.

First, it's beneficial to sketch the base function. Start by plotting points you know, like where \(f(x) = \log x\) crosses \((1,0)\). Then, observe the behavior as \(x\) nears zero and extends far beyond one. This gives you the curving pattern typical of logarithms.

Second, apply any transformations:

• Shift the graph by moving each point according to the transformation rules—like in the shift experienced by \(g(x) = \log(x-2) + 1\). Move every plotted point 2 units right and 1 unit up.
• Draw the new graph based on these movements, ensuring the characteristic curve remains accurate.

By mastering graphing techniques, you gain the ability to manipulate and understand how the algebraic transformations affect a graph visually. This is particularly handy when comparing functions, such as in our exercise, where the visual differences speak volumes about the transformations applied.