Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=2^{x}+3$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 2^{x} + 3\) is an increasing curve and has a horizontal asymptote at \(y = 3\).

Step 1: Constructing the table of values

Pick several values for \(x\) and calculate the corresponding \(f(x)\) values. These could be, for example, -3, -2, -1, 0, 1, 2, 3.

Step 2: Sketching the graph

Use the table of values to sketch the graph. Plot the points on the graph and then connect them to form a curve.

Step 3: Identifying the asymptotes

Asymptotes are lines that the graph approaches but never crosses. In the case of this function, it has a horizontal asymptote at \(y = 3\) as \(x\) approaches negative infinity. There is no vertical asymptote because the function is defined for all real numbers.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in mathematics that helps us visualize how a function behaves. For the function given in the problem, \(f(x) = 2^x + 3\), the graph shows us how the values of \(f(x)\) change as \(x\) varies. To graph a function like this, we typically start by creating a list of values for \(x\) and their corresponding \(f(x)\) values. This is known as a function table, and it helps us see exact points that we will plot on a graph.

After we have several ordered pairs from our function table, we plot these on a coordinate plane. Once the points are plotted, the next step is to connect them smoothly. This results in the curve we call the graph of the function. In the case of \(2^x + 3\), the graph will be an upward-sloping curve due to the nature of exponential growth. Exponential functions grow rapidly as \(x\) increases and flatten out as \(x\) becomes very negative.

Remember, graphing is very visual and seeing how the function behaves at various points can give insights into its properties and behavior over different intervals.

Horizontal Asymptotes

Horizontal asymptotes are lines that the graph of a function approaches as the input values (\(x\)) become extremely large or small. They give us an idea of the behavior of the function at the 'ends' of the graph, making them crucial for understanding long-term behavior. For instance, in the given function \(f(x) = 2^x + 3\), there is a horizontal asymptote at \(y = 3\).

This is because, as \(x\) approaches negative infinity (very large negative values), \(2^x\) approaches zero. Since adding 3 is a constant shift upward, the graph approaches \(y = 3\) but does not cross it. However, unlike some asymptotes that dictate the behavior across the entire graph, this horizontal asymptote only influences the graph as \(x\) becomes very negative. The graph definitely crosses \(y = 3\) as \(x\) increases from zero onwards. Understanding horizontal asymptotes helps us predict and describe the end-behavior of functions.

Function Table

A function table is a structured way to see how changes in \(x\) affect the function value \(f(x)\). This table forms the backbone of your graph plot points. For creating a function table with \(f(x) = 2^x + 3\), choose several \(x\) values such as -3, -2, -1, 0, 1, 2, and 3. Solving for \(f(x)\) gives you the precise points to plot.

Here's how you compute these values: plug each \(x\) into \(2^x + 3\) to get \(f(x)\). For example, if \(x = 1\), then \(f(1) = 2^1 + 3 = 5\). Continue this for other values of \(x\) and write them down in your function table.

Using the table, you can easily create a plot as you now know the exact coordinates to plot on the graph. This helps to visualize whether the function is increasing, decreasing, or undergoes any transformations. It's a useful tool for ensuring accuracy when sketching the graph manually without a calculator.