Problem 5 - Algebra and Trigonometry

Question

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ f(x)=2^{x} $$

Step-by-Step Solution

Answer

Plot \((-2, \frac{1}{4}), (-1, \frac{1}{2}), (0, 1), (1, 2), (2, 4)\) and draw the curve.

Step 1: Choose Values for x

Start by selecting a few values for \( x \). A good range might be from -2 to 2. So, we will use \( x = -2, -1, 0, 1, \) and \( 2 \).

Step 2: Calculate f(x) for Each Chosen x

Compute \( f(x) = 2^x \) for each chosen \( x \). - When \( x = -2, f(x) = 2^{-2} = \frac{1}{4} \).- When \( x = -1, f(x) = 2^{-1} = \frac{1}{2} \).- When \( x = 0, f(x) = 2^0 = 1 \).- When \( x = 1, f(x) = 2^1 = 2 \).- When \( x = 2, f(x) = 2^2 = 4 \).

Step 3: Create a Table of Values

Organize the results into a table:\[\begin{array}{c|c}x & f(x) \\hline-2 & \frac{1}{4} \-1 & \frac{1}{2} \0 & 1 \1 & 2 \2 & 4 \\end{array}\]

Step 4: Sketch the Graph

On graph paper or using graphing software, plot the points from the table: - Plot \((-2, \frac{1}{4})\)- Plot \((-1, \frac{1}{2})\)- Plot \((0, 1)\)- Plot \((1, 2)\)- Plot \((2, 4)\)Then, draw a smooth curve through these points to represent the function \( f(x) = 2^x \). Notice the exponential growth as \( x \) increases.

Key Concepts

Graphing Functions

Graphs visually represent mathematical functions. They help us understand how a function behaves across different values of the variable. In the example provided, we graph the exponential function \(f(x) = 2^x\). Graphing starts by plotting data points on a coordinate plane, based on the ordered pairs from our calculated results. These pairs consist of an input \(x\) and the corresponding output \(f(x)\) value.

To sketch a graph, use:

• A coordinate plane: Horizontal axis for \(x\) (input) and vertical axis for \(f(x)\) (output).
• Data points: Place these calculated pairs on the plane.
• Connect: Draw a smooth line or curve through the points to outline the function's behavior.

For exponential functions like \(2^x\), notice how quickly the graph rises as \(x\) increases! This rapid increase is a hallmark of exponential growth.

Table of Values

To graph a function accurately, creating a table of values is essential. This table is a simple way to organize different inputs \(x\) and their corresponding outputs \(f(x)\). For the function \(f(x) = 2^x\), you begin by selecting a set of \(x\) values. Each chosen \(x\) is inputted into the function to calculate \(f(x)\). The outcomes help in understanding how the function behaves.

Steps to create a table of values include:

• Select \(x\) inputs: Choose a range that captures the behavior of the function. Commonly, \(x = -2, -1, 0, 1, 2\) works well for simple examples.
• Calculate \(f(x)\): Compute the output using the function formula, like \(f(x) = 2^x\).
• Organize: Match each input with its output for clarity.

This organized approach helps plot the graph seamlessly and shows the relationship between \(x\) and \(f(x)\).

Exponential Growth

The concept of exponential growth is crucial in understanding functions like \(f(x) = 2^x\). An exponential function grows by a constant multiplicative rate rather than a constant additive rate. This means as \(x\) increases, \(f(x)\) grows much faster compared to linear growth. In an exponential function, the base number (which is \(2\) in this example) plays a pivotal role.

Key features of exponential growth:

• Doubling rate: Each increase in \(x\) results in the doubling of \(f(x)\), showing a rapid rise.
• Small beginnings, big endings: Close to \(x = 0\), the values of \(f(x)\) start small but escalate quickly.
• Graph shape: Exponential graphs have a characteristic J-shape, steeply rising as \(x\) becomes positive.

Understanding this pattern is vital as it models many real-life situations, such as population growth or interest compounding.