Problem 26 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=5+2 e^{x}$$

Step-by-Step Solution

Answer

The graph of the function has a y-intercept at \(y = 7\), and it rises indefinitely as \(x\) increases, and approaches a horizontal line as \(x\) decreases. Actual y-values depend on specific x-values taken, and should be accurately calculated for precise sketching.

Step 1: Determine the y-intercept

The y-intercept of the graph will occur when \(x = 0\). Therefore, substituting \(x = 0\) into the function, the y-intercept can be found as \(f(0) = 5 + 2 * e^0 = 5 + 2 = 7\).

Step 2: Choose a set of x-values

Now, choose some x-values to determine the shape of the function. Note that since it is an exponential function, it is useful to choose both negative and positive x-values. Let's choose -2, -1, 0, 1, 2.

Step 3: Compute corresponding y-values

The y-values will be found by substituting each chosen x-value into the function, the y-values will be -2: \(f(-2) = 5 + 2 * e^{-2}\), -1: \(f(-1) = 5 + 2 * e^{-1}\), 0: \(f(0) = 7\) (from the previous step), 1: \(f(1) = 5+2 * e^{1}\), and 2: \(f(2) = 5+2 * e^{2}\) approximately.

Step 4: Sketch the graph

Now use the computed points to sketch the graph, keeping in mind that as an exponential function, the graph will approach a horizontal asymptote as \(x\) approaches negative infinity, and it will rise indefinitely as \(x\) approaches positive infinity.

Key Concepts

Graphing Functions

Graphing functions can be a fun and intuitive process, especially with exponential functions. To graph a function, follow these steps:

• First, understand the general shape of the function you are dealing with. For exponential functions like \(f(x) = 5 + 2e^x\), the graph tends to rise steeply as \(x\) increases and flattens towards a horizontal asymptote as \(x\) decreases.
• Next, determine critical points such as the y-intercept by setting \(x = 0\).
• Then, evaluate the function at several x-values, both positive and negative, to get a sense of the graph's behavior and shape.
• Finally, plot these points and sketch a smooth curve through them, keeping the growth pattern of the function in mind.

By visualizing each part of this process, graphing functions becomes a powerful tool for understanding exponential relationships.

Y-intercept

The y-intercept of a graph is crucial because it tells us where the function crosses the y-axis. For the function \(f(x) = 5 + 2e^x\), the y-intercept is found by substituting \(x = 0\). This means you look at \(f(0) = 5 + 2e^0\).

• Since \(e^0 = 1\), you simply calculate \(f(0) = 5 + 2 \times 1 = 7\).
• This tells us that when \(x = 0\), \(f(x) = 7\), so the y-intercept is the point (0, 7).

The y-intercept is useful for anchoring the graph and providing a reference point for plotting other values.

Horizontal Asymptote

A horizontal asymptote refers to a line that a graph approaches but never quite reaches as \(x\) moves towards positive or negative infinity. For an exponential function like \(f(x) = 5 + 2e^x\), the constant term 5 identifies the horizontal asymptote. This is because as \(x\) approaches negative infinity, \(2e^x\) approaches zero.

• Thus, the horizontal asymptote is the line \(y = 5\).
• This means that as \(x\) becomes very negative, the function approaches a value of 5, but never dips below.

Recognizing the horizontal asymptote helps to frame the graph and gives an understanding of the function's long-term behavior.

Exponential Growth

Exponential growth is a hallmark of functions like \(f(x) = 5 + 2e^x\). These functions are characterized by rapid increases in value as \(x\) becomes larger. This happens because the base of the exponential function, in this case \(e\), is greater than one.

• As \(x\) increases, \(e^x\) grows quickly, causing \(f(x)\) to increase without bound.
• This is evident in the graph, where the curve steeply rises as it moves rightward.

Understanding exponential growth is crucial in real-world applications where populations, investments, or other quantities grow at a constant rate, highlighting the significance of exponential functions in various fields.