Problem 6 - Algebra and Trigonometry

Question

5–10 ? Sketch the graph of the function by making a table of values. Use a calculator if necessary. $$ g(x)=8^{x} $$

Step-by-Step Solution

Answer

The graph of \( g(x) = 8^x \) is an exponential curve, rising rapidly when \( x \) is positive.

Step 1: Understanding the Function

The function given is an exponential function, written as \( g(x) = 8^x \). Exponential functions have a constant base raised to a variable exponent. The base \( 8 \) means the function will grow rapidly as \( x \) increases.

Step 2: Create a Table of Values

Let's pick several values for \( x \) to calculate \( g(x) \). Common choices are \( x = -2, -1, 0, 1, 2, 3 \), giving us a range of values to observe the function's behavior. Compute \( g(x) \) for each \( x \):- \( x = -2 \): \( g(x) = 8^{-2} = \frac{1}{64} \)- \( x = -1 \): \( g(x) = 8^{-1} = \frac{1}{8} \)- \( x = 0 \): \( g(x) = 8^{0} = 1 \)- \( x = 1 \): \( g(x) = 8^{1} = 8 \)- \( x = 2 \): \( g(x) = 8^{2} = 64 \)- \( x = 3 \): \( g(x) = 8^{3} = 512 \)

Step 3: Sketch the Graph

Based on the table of values from Step 2, plot the points \((-2, \frac{1}{64})\), \((-1, \frac{1}{8})\), \((0, 1)\), \((1, 8)\), \((2, 64)\), and \((3, 512)\) on a coordinate plane. The graph will rapidly rise on the right side as \( x \) increases and will get very close to the x-axis on the left as \(x\) becomes negative. Connect the points smoothly to form an exponential curve.

Key Concepts

Exponential Function

An exponential function is a mathematical expression that features a constant base raised to a variable exponent. In our case, the function is given as \(g(x) = 8^x\). Here, the base is \(8\) which means that as \(x\) changes, the value of \(g(x)\) grows or shrinks exponentially. This makes exponential functions very interesting and powerful, especially in modeling situations where growth occurs at a rapid rate.
Exponential functions are characterized by their unique growth patterns. Unlike linear functions, which increase at a constant rate, exponential functions grow faster and faster as \(x\) increases.
They form curves on a graph that can rise or fall sharply, depending on the base and the sign of the exponent.

• If the base is greater than \(1\), as the case we have with \(8\), the function will grow exponentially as \(x\) increases, demonstrating exponential growth.
• Conversely, if the exponent is negative, the function will shrink rapidly as \(x\) increases.

Understanding these properties is crucial to graphing exponential functions effectively.

Table of Values

A table of values is an essential tool in graph sketching, especially for unfamiliar functions like \(g(x) = 8^x\). Constructing a table of values allows us to see the behavior of the function across different inputs of \(x\).
When creating a table of values, you choose several input values for \(x\), calculate the corresponding \(g(x)\) values, and document them in a simple and organized manner:

• \(x = -2\): \(g(x) = \frac{1}{64}\)
• \(x = -1\): \(g(x) = \frac{1}{8}\)
• \(x = 0\): \(g(x) = 1\)
• \(x = 1\): \(g(x) = 8\)
• \(x = 2\): \(g(x) = 64\)
• \(x = 3\): \(g(x) = 512\)

This organized set of points gives us solid ground for plotting and sketching the graph. Each calculated point represents a specific point on the curve in the coordinate plane, paving the way for understanding the function's behavior at those points and beyond.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot functions by marking points that correspond to specific input-output pairs. This plane consists of two axes: the horizontal \(x\)-axis and the vertical \(y\)-axis.
In our example, each point in the table of values for the function \(g(x) = 8^x\) is plotted on this plane by following these guidelines:

• Select an \(x\) value from your table.
• Locate this \(x\) value on the horizontal axis.
• Find \(g(x)\), the computed value, on the vertical axis.
• Mark the point where the \(x\) and \(g(x)\) intersect.

As plotted points from the values \((-2, \frac{1}{64})\), \((-1, \frac{1}{8})\), \((0, 1)\), \((1, 8)\), \((2, 64)\), and \((3, 512)\), these form the outline of the graph for the function \(g(x) = 8^x\). The relationship on the coordinate plane visually displays the exponential nature of the function, showing rapid growth for increasing positive \(x\) values, and approaching zero as \(x\) decreases.

Exponential Growth

Exponential growth refers to the process where quantities like populations or values of an investment increase at a rate proportional to their current size. This distinguishes it from linear growth, where additions are constant, not dependent on existing quantities.
In mathematical terms, exponential growth is aptly demonstrated by functions like \(g(x) = 8^x\). As \(x\) increases, \(g(x)\) doesn't grow linearly but rather multiplies at an increasing rate. Hence, even a slight increase in \(x\) can result in a significant increment to the function's value.
In real-world applications, exponential growth is often found in areas such as finance (interest calculations), populations (populations doubling over time), or any scenario where a quantity multiplies rapidly. The exponential function of our exercise elegantly models this behavior, offering a graphical representation that shows explosive growth as you move right along the \(x\)-axis.
Recognizing the pattern of exponential growth is pivotal, as it helps in predicting trends and understanding potential outcomes in practical scenarios. Understanding these can guide in making informed decisions based on mathematical projections.