Problem 28 - 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. \(g(x)=2^{x}+2\)

Step-by-Step Solution

Answer

The graph of \(g(x)=2^{x}+2\) is the same as the graph of \(f(x)=2^{x}\), but shifted upwards by 2 units. The point (0,1) on \(f(x)\) moves to (0,3) on \(g(x)\).

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

Start by drawing the graph of \(f(x) = 2^{x}\). This is an exponential function with base 2. The graph passes through (0,1) because anything to the power of zero is one, and it goes infinitely up to the right while infinitely approaching zero to the left.

Step 2: Understand the transformation

The given function \(g(x) = 2^{x} + 2\) represents a vertical shift of \(f(x)\) upwards by 2 units. In general, if you add or subtract from a function, it shifts the graph vertically. In this case, adding 2 to \(2^{x}\) means every point on the \(f(x) = 2^{x}\) graph will move up by 2 units.

Step 3: Apply the transformation and graph \(g(x)\)

Apply the transformation to some key points of \(f(x)\) to get the same points for \(g(x)\). For example, the point (0, 1) on \(f(x)\) will become (0, 3) on \(g(x)\). Similarly, the point where \(f(x)\) intersects the y-axis at (0,1) will move to (0,3). After transforming every point, connect them to draw the graph.

Step 4: Confirm the graph with a graphing utility (optional)

If you want to confirm your hand-drawn graph or if the exercise allows it, use a graphing utility to graph \(g(x)\). This step can provide a handy visual, especially for those more complicated transformations.

Key Concepts

Exponential Function Transformation

Understanding how to transform an exponential function is foundational in math. An exponential function is of the form f(x) = a^x, where a is a constant and x represents the exponent. Transformations involve shifting, stretching, compressing, or reflecting the basic graph. The graph of f(x) = 2^x is the starting point, featuring a curve that rapidly increases as x moves to the right and approaches zero as x moves to the left.

Transforming this function could involve various operations such as f(x) = a^x + k or f(x) = a^(x + h) and others. Each of these manipulates the graph in a predictable way; adding or subtracting a value, for example, moves the graph up or down, while changing the base a affects the rate of growth or decay.

Vertical Shift

In the case of g(x) = 2^x + 2, we're looking at a 'vertical shift'. This is a type of transformation where the graph maintains its shape but is moved either up or down along the y-axis. For a positive value of k in f(x) = a^x + k, the entire graph shifts up by k units. Conversely, if k is negative, the shift is downward.

To visually apply a vertical shift, it's useful to consider key points. For instance, if the point (0, 1) is on the function f(x), adding 2 to the function transforms this point to (0, 3) on g(x). By shifting each point individually and then re-drawing the graph, we can visualize the new function with ease. A vertical shift does not affect the x-values of any points on the graph, only the y-values.

Graphing Calculator Utility

Graphing calculators are incredibly beneficial tools when studying exponential functions and their transformations. You can visualize the graph of a function, like g(x) = 2^x + 2, much more quickly than by sketching by hand, which is particularly useful when confirming your understanding of transformations. Such utilities often provide interactive features, allowing you to tweak the function parameters and observe the resulting changes to the graph in real-time.

When working with harder-to-visualize transformations, graphing utilities become even more invaluable. They offer a precise and immediate representation of the graph, which helps you verify the accuracy of your manual transformations. These utilities are also essential teaching aids, as they help students grasp the immediate impact of each transformation on the overall shape and position of the graph.