Problem 30 - 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. \(h(x)=2^{x+2}-1\)

Step-by-Step Solution

Answer

The graph of \(h(x)=2^{x+2}-1\) is the same as the graph of the original function \(f(x)=2^{x}\) but shifted two units to the left and one unit down. This can be validated using a graphing utility.

Step 1: Graph the Original Function

Start with graphing the original function \(f(x)=2^{x}\) on a grid or graph paper for different values of \(x\), ranging from -2 to 2.

Step 2: Understand the Transformations

Next, examine how \(h(x)=2^{x+2}-1\) transforms from \(f(x)=2^{x}\). The \(+2\) in the exponent represents a shift two units to the left, while the \(-1\) translates the graph down one unit.

Step 3: Apply the Transformations and Graph \(h(x)\)

According to these transformations, graph \(h(x)=2^{x+2}-1\) by shifting every point on the graph of \(f(x)=2^{x}\) two units to the left and one unit down.

Step 4: Confirm with a Graphing Utility

Finally, validate your hand-drawn graph using a graphing utility like Grapher or Desmos. Input the function \(h(x)=2^{x+2}-1\) and check if the result matches your hand-drawn graph.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They often have the form \(f(x) = a^{x}\), where \(a\) is a positive constant. These functions show rapid growth or decay and are used in various real-world applications, like population growth or radioactive decay. For the function \(f(x) = 2^x\), the base is 2, and as \(x\) increases, the function values double, showing exponential growth.
Compared to polynomial functions, exponential functions rise or fall more quickly. It's important to get comfortable with their basic graph shape:

• The graph passes through the point (0, 1), since any number to the zeroth power is 1.
• The graph is asymptotic to the x-axis; that is, it approaches the x-axis but never touches it.
• When \(a > 1\), like in \(2^x\), the graph moves upwards as \(x\) increases.

Understanding these characteristics will help in transforming these graphs successfully.

Horizontal Shift

A horizontal shift moves the graph of a function left or right along the x-axis. In the function \(h(x) = 2^{x+2} - 1\), the \(+2\) inside the exponent indicates a shift to the left. Why left? Because adding inside the function's formula moves graphs in the opposite horizontal direction.
Think of it like this: for every \(x\), you start altering the value earlier by 2 units. It can be a useful representation for time shifts or delays in changes over time.
To apply this transformation on the graph of \(f(x) = 2^x\):

• Move all points horizontally 2 units to the left.
• Each x-coordinate becomes \(x - 2\).

Mastering horizontal shifts helps you understand how slight modifications alter the entire graph's behavior.

Vertical Translation

Vertical translation shifts a graph up or down along the y-axis. For the function \(h(x) = 2^{x+2} - 1\), the \