Problem 39 - 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. $$f(x)=\log _{2}(x+4)$$

Step-by-Step Solution

Answer

Shift the graph of \(g(x) = \log_{2}x\) 4 units left to get \(f(x) = \log_{2}(x+4)\).

Step 1: Identify Changes to Parent Function

The given function is \(f(x) = \log_{2}(x+4)\). This function is based on the parent function \(g(x) = \log_{2}x\). Notice that the argument of the logarithm, \(x\), is replaced with \(x+4\).

Step 2: Identify Horizontal Shift

The replacement of \(x\) with \(x+4\) indicates a horizontal shift. In the context of transformations, \(x+4\) means the graph of \(g(x) = \log_{2}x\) will shift 4 units to the left. This is because you set the expression \(x+4\) equal to zero to find the shift: \(x+4=0 \Rightarrow x=-4\).

Step 3: Graph the Transformed Function

Start by sketching the basic graph of \(g(x) = \log_{2}x\), which has a vertical asymptote at \(x = 0\) and passes through the point \((1,0)\). For \(f(x) = \log_{2}(x+4)\), shift every point on this graph 4 units to the left, causing the vertical asymptote to move from \(x=0\) to \(x=-4\). The point \((1,0)\) shifts to \((-3,0)\). As a result, the entire graph moves leftwards.

Key Concepts

Parent function

The concept of a parent function is like the foundation for a building. It provides the basic shape and behavior of a graph. In mathematics, a parent function is the simplest form of a set of functions that form a family. For logarithmic functions, the parent function is generally expressed as

• \(g(x) = \log_{b}x\),

where \(b\) is the base of the logarithm. This function has a distinctive shape, featuring a vertical asymptote on the y-axis, a domain of \((0, \infty)\), and a range of

• \((-\infty, \infty)\).

For

• the base 2 logarithmic function \(g(x) = \log_{2}x\),

the graph passes through the point

• \((1, 0)\)

, among others. \(g(x) = \log_{2}x\) is like "the blueprint" for all other transformations that may follow. By understanding the shape of this parent function, changes like shifts and stretches become easier to visualize.

Horizontal shift

A horizontal shift refers to moving the graph of a function left or right along the x-axis. It's as if you're sliding the graph to a new position without altering its shape. In the exercise for the function

• \(f(x) = \log_{2}(x+4)\)

the argument of the logarithm includes \(x+4\). This indicates a shift.

To determine the direction and magnitude of the shift, set the "parent" argument, \(x\), equal to the transformed argument, \(x+4\), and solve for the shift:\[x+4 = 0 \Rightarrow x = -4\]This shows that each point on the parent graph will move 4 units to the left.

Here's a quick way to remember:

• When you see \(x+c\), shift the graph \(c\) units left.
• Conversely, \(x-c\) means a shift \(c\) units right.

By applying this shift, the entire graph of \(f(x) = \log_{2}(x)\) is relocated, with all its unique features, like asymptotes and intercepts, moving accordingly.

Logarithmic function

Logarithmic functions uniquely turn exponential relationships into additive ones. They're used widely in science and engineering to compress data ranges because they map from an infinite domain to a much smaller range. The general form of a logarithmic function is

• \(f(x) = \log_{b}x\)

, where \(b\) is the base of the logarithm.

These functions have several properties:

• They pass through the point \((1, 0)\) because \(\log_{b}1 = 0\), for any base \(b\).
• The graph approaches a vertical asymptote as \(x\) approaches zero from the right, meaning \(x > 0\).
• Increasing base \(b\) makes the graph steeper.

For the function

• \(f(x) = \log_{2}(x+4)\),

it follows the basic behavior of \(\log_{2}x\) but with changes derived from transformations.

These transformations are crucial in plotting the accurate graph on the coordinate plane. It's essential to understand these behaviors as they help in predicting how each function will behave under different conditions.