Problem 34 - College Algebra

Question

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=\frac{1}{2} \cdot 2^{x}\)

Step-by-Step Solution

Answer

The graph of \(g(x)=\frac{1}{2} \cdot 2^{x}\) can be obtained by a vertical shrink of the graph of \(f(x)=2^{x}\) by a factor of \(\frac{1}{2}\).

Step 1: Graph the function \(f(x)=2^{x}\)

Begin by creating an x-y graph. Pick a few values for x, such as -2, -1, 0, 1, 2 and compute corresponding values for \(2^{x}\). For example, when \(x=-2, 2^{-2}=\frac{1}{4}\), and when \(x=0, 2^{0}=1\). Plot these points on the graph to create the plot for \(f(x)\).

Step 2: Understand the transformation

Looking at \(g(x)=\frac{1}{2} \cdot 2^{x}\), we notice it's \(f(x)\) scaled by a factor of \(\frac{1}{2}\) in the y-direction. This means every y-coordinate of \(f(x)\) is scaled by a factor of \(\frac{1}{2}\) to give the graph of \(g(x)\).

Step 3: Graph \(g(x)=\frac{1}{2} \cdot 2^{x}\)

Use the same x-values as Step 1, but scale the y-values by a factor of \(\frac{1}{2}\) to create the plot for \(g(x)\). So, if for a given x, the y value on function \(f(x)\) was y, then for function \(g(x)\), it will be \(\frac{1}{2} * y\). Plot these points on the graph.

Step 4: Confirm the graph using a graphing utility

Enter the functions \(f(x)=2^{x}\) and \(g(x)=\frac{1}{2} \cdot 2^{x}\) into a graphing utility and observe that the graph of \(g(x)\) is indeed a vertical shrink of the graph of \(f(x)\) by a factor of \(1/2\). This confirms the hand-drawn graphs.

Key Concepts

Graph Transformations of Exponential Functions

When dealing with functions like the exponential function, graph transformations allow us to see how the graph changes when certain operations are applied. In our exercise, we started by looking at the graph of the basic exponential function,

• The function: \(f(x) = 2^x\) is our starting point.
• To transform this graph into the graph of \(g(x) = \frac{1}{2} \cdot 2^x\), we apply transformations.

The main transformation here is the vertical scaling, which we will explore shortly. Graph transformations can include shifting, reflecting, and scaling the graph. Understanding the basic graph helps stakeholders identify and plot transformed versions on the same set of axes.
The process begins by plotting points for several values of \(x\) for the original function. This serves as a foundation to understand how the transformation affects each point. Graph transformations, especially for exponential functions, vividly illustrate how changes in formulas translate visually into different graph shapes.

Vertical Scaling of Exponential Graphs

Vertical scaling involves multiplying the function's output by a constant, affecting the graph's appearance along the y-axis. In our example, the function \(g(x) = \frac{1}{2} \cdot 2^x\) is a vertically scaled version of \(f(x) = 2^x\).

• Vertical scaling by \(\frac{1}{2}\) causes the graph to shrink towards the x-axis, reducing each point's y-value in half.
• If \(f(x) = 2^x\) results in y-values like 1, 2, or 4 for x-values 0, 1, and 2 respectively, then \(g(x)\) results in \(\frac{1}{2}\), 1, or 2 after scaling.

This type of transformation does not affect the x-values of the graph, meaning the horizontal structure remains consistent while the vertical spread changes. Vertical scaling is an essential transformation as it adjusts the rate at which an exponential function grows or shrinks, affecting its steepness or flatness.

Coordinate Plotting for Graph Clarity

Coordinate plotting is a valuable tool for visually presenting functions accurately. It ensures we correctly capture the transformation effects, like vertical scaling, across all plotted points. In our scenario involving \(f(x) = 2^x\) and \(g(x) = \frac{1}{2} \cdot 2^x\), plotting is performed by first determining the points for the base function.

• Start by selecting key x-values, such as -2, -1, 0, 1, and 2.
• Calculate y-values using the function for each x-value.
• Plot these points as precisely as possible on a graph for a clear visual representation.

For the second function, repeat the process using the scaled y-values. This ensures points on the new graph accurately reflect the vertical scaling. Consistently using the same x-values helps compare transformations against the untransformed graph, enhancing understanding of how scaling affects graph behavior and appearance visually.