Problem 9 - Algebra and Trigonometry

Question

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ g(x)=3 e^{x} $$

Step-by-Step Solution

Answer

Graph the exponential function by calculating values for selected x-values and plotting them.

Step 1: Understanding the Function

The given function is \( g(x) = 3e^x \). This is an exponential function where \( e \) is the base, also known as Euler's number, approximately equal to 2.718. The 3 in front of \( e^x \) is a coefficient that vertically stretches the graph.

Step 2: Select x-values for the Table

Choose a few x-values to plot the function. Common choices are \( x = -2, -1, 0, 1, 2 \). This will give us a range of points that show the general shape of the graph.

Step 3: Calculate y-values

Substitute the chosen x-values into the function to find the corresponding y-values. Calculate the following:For \( x = -2 \), \( g(-2) = 3e^{-2} \).For \( x = -1 \), \( g(-1) = 3e^{-1} \).For \( x = 0 \), \( g(0) = 3e^{0} = 3 \).For \( x = 1 \), \( g(1) = 3e^{1} \).For \( x = 2 \), \( g(2) = 3e^{2} \).

Step 4: Calculate Specific Values

Using a calculator:\( e^{-2} \approx 0.1353 \), so \( g(-2) \approx 3 imes 0.1353 = 0.4059 \)\( e^{-1} \approx 0.3679 \), so \( g(-1) \approx 3 imes 0.3679 = 1.1037 \)\( e^0 = 1 \), so \( g(0) = 3 \times 1 = 3 \)\( e^1 \approx 2.718 \), so \( g(1) \approx 3 imes 2.718 = 8.154 \)\( e^2 \approx 7.389 \), so \( g(2) \approx 3 imes 7.389 = 22.167 \)

Step 5: Sketch the Graph

Plot the points \((x, g(x))\) from the table: \((-2, 0.4059), (-1, 1.1037), (0, 3), (1, 8.154), (2, 22.167)\). Connect these points smoothly, noting that the graph is increasing and exponential. The curve should start near the x-axis for very negative x-values and rapidly increase as x becomes positive.

Key Concepts

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the form \( f(x) = ab^x \), \( a \) is a constant coefficient, \( b \) is the base and \( x \) is the variable exponent. Exponential functions are characterized by their rapid growth or decay as the input \( x \) increases or decreases.
They play a crucial role in various fields such as finance, science, and engineering, often modeling growth processes like population increase or radioactive decay. In the function \( g(x) = 3e^x \), \( e^x \) indicates that the base is Euler's number \( e \), and the coefficient 3 stretches the function vertically. As \( x \) grows, \( 3e^x \) increases at a rate proportional to its current value, highlighting the core property of exponential functions.

Euler's Number

Euler's number, commonly denoted as \( e \), is approximately 2.718. It is an irrational number, meaning it cannot be expressed exactly as a simple fraction. Euler's number is fundamental in mathematics because it serves as the base of natural logarithms and has unique properties that make it crucial in calculus.
It emerges naturally in processes that exhibit continuous growth, such as compound interest calculations. When used as the base of an exponential function, \( e \) provides a standard form of exponential growth called the natural exponential function. In the function \( g(x) = 3e^x \), \( e \) ensures that the growth of the function is smooth and continuous, allowing mathematicians and scientists to predict changes over time efficiently.

Table of Values

Creating a table of values is a foundational step in graphing functions. The table lists input values (usually along a range of interest) and their corresponding output values from the function. For \( g(x) = 3e^x \), we choose specific values of \( x \) such as -2, -1, 0, 1, and 2. By substituting these into the function, we calculate the respective \( g(x) \) values. This method helps visualize how the function behaves and is especially useful for identifying the shape of the graph.
For instance, when \( x = -2 \), the output is calculated as \( g(-2) = 3e^{-2} \approx 0.4059 \). Continuing this for other values, such as \( x = 0 \) giving \( g(0) = 3 \), the table of values we create serves as a guide for plotting the points on a graph. This plotted graph illustrates the exponential nature of \( g(x) \), rising sharply as \( x \) increases.