Problem 25 - 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. \(g(x)=2^{x+1}\)

Step-by-Step Solution

Answer

The function \(g(x)=2^{x+1}\) is a transformation of the base function \(f(x)=2^{x}\). This transformation corresponds to a shift one unit to the left in the x-direction. The key points for graphing \(g(x)=2^{x+1}\) will be \((-3,0.25), (-2,0.5), (-1,1), (0,2), (1,4)\).

Step 1: Understanding the Base Function

Begin by plotting points for the base function \(f(x)=2^{x}\). The key points are typically where \(x\) is -2, -1, 0, 1, and 2 to get a wide view of what the curve looks like.

Step 2: Plotting the Base Function

Using the points from Step 1, plot the graph of \(f(x)=2^{x}\) on a grid. Make sure to label your axes and points clearly. For \(f(x)=2^x\), points will be \((-2,0.25), (-1,0.5), (0,1), (1,2), (2,4)\)

Step 3: Understanding the Transformation

The function \(g(x)=2^{x+1}\) is a transformation of the base function where 1 is added to each x-value. This is called a horizontal shift. In this case, it’s a shift one unit to the left.

Step 4: Plotting the Transformed Function

Now graph the transformation by shifting all points from the base function one step to the left. The points for \(g(x)=2^{x+1}\) will be \((-3,0.25), (-2,0.5), (-1,1), (0,2), (1,4)\). Label these points clearly on your grid.

Step 5: Confirming the Result with a Graphic Utility (Optional)

This step, though optional, can be done with any graphic calculator or online tool. By inputting the function \(g(x)=2^{x+1}\) into a graphing utility, it will show a graph that should look similar to your hand-drawn graph. This serves as a double check. This step is highly encouraged, as it helps to ensure accuracy and understanding of the concept.

Key Concepts

Base Function

When learning about graphing exponential functions, one must first become familiar with the concept of a base function. A base function serves as the starting point for understanding transformations. In the context of exponential functions, a base function has the form f(x) = a^x, where a is a constant, and x is the exponent.

For instance, the function f(x) = 2^x is an exponential base function with base 2. In this function, as the exponent x increases, the output value of the function rapidly grows, reflecting exponential growth. To visualize this concept, one plots key points such as (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), and (2, 4). These points help to capture the curve's behavior as it moves either towards zero (as x becomes large and negative) or becomes very large (as x becomes positive).

Function Transformation

Function transformation involves taking a base function and applying specific changes to its formula that alter its graph in predictable ways. Transformations include shifting, stretching, compressing, and reflecting the graph. These transformations can occur vertically or horizontally, affecting either the x-values or the y-values of the points that define the function's curve.

In the exercise, the transformation consists of adding 1 to each x-value of the base function f(x) = 2^x, resulting in the new function g(x) = 2^{x+1}. This sort of transformation affects the horizontal positioning of the graph, where each point on the base function's graph moves according to the change applied to the x-value. Understanding how these transformations affect the graph is key to mastering the art of graphing functions.

Horizontal Shift

A horizontal shift in the context of function transformation refers to moving the entire graph of a function to the left or the right along the horizontal axis. This shift is often a result of adding or subtracting a constant to the x-variable within the function's formula.

For example, when one goes from f(x) = 2^x to g(x) = 2^{x+1}, there is a horizontal shift involved. By adding 1 inside the exponent, each point on the graph of f(x) moves one unit to the left to form the graph of g(x). It's important to note such transformations do not change the shape of the graph but simply reposition it. By visualizing this horizontal shift, one can accurately graph the transformed function.

Exponential Growth

Exponential growth is a specific way in which certain quantities increase over time. In the context of graphing functions, when we talk about exponential growth, we are referring to how the value of a function, such as f(x) = 2^x, increases as the exponent x gets larger. Exponential functions grow at a rate proportional to their value, which means the higher the function's output, the faster it grows.

This rapid increase is visually evident on a graph as the curve steepens the further right it goes (where x is positive). Conversely, as x becomes negative, the function approaches zero but never reaches it, reflecting a horizontal asymptote. Understanding this behavior is essential for interpreting graphs of exponential functions and predicting their growth patterns in various applications.