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Chapter 5: Problem 10

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

Short Answer

Expert verified
Plot points for \( x = -2, -1, 0, 1, 2 \) and sketch the exponential curve.

Step by step solution

01

Create a Table of Values

To sketch the graph of the function \( g(x) = 8^x \), we start by creating a table of input values \( x \) and their corresponding \( g(x) \) values. Select a range of \( x \) values, for example, \(-2, -1, 0, 1, 2\). We'll calculate \( g(x) \) for each of these values.
02

Calculate Values for Selected x

Now, calculate \( g(x) \) for each selected \( x \):- For \( x = -2 \), \( g(-2) = 8^{-2} = \frac{1}{64} \approx 0.0156 \)- For \( x = -1 \), \( g(-1) = 8^{-1} = \frac{1}{8} = 0.125 \)- For \( x = 0 \), \( g(0) = 8^0 = 1 \)- For \( x = 1 \), \( g(1) = 8^1 = 8 \)- For \( x = 2 \), \( g(2) = 8^2 = 64 \)
03

Plot the Points on a Graph

Using the table of values calculated, plot the points \((-2, 0.0156)\), \((-1, 0.125)\), \((0, 1)\), \((1, 8)\), and \((2, 64)\) on a graph. These points show how the function behaves for the given \( x \) values.
04

Draw the Graph

Connect the plotted points with a smooth curve to form the graph of \( g(x) = 8^x \). The graph should approach the x-axis as \( x \) becomes more negative and rise steeply as \( x \) increases. This exponential growth shows the rapid increase in the value of \( g(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth
Exponential growth describes a phenomenon where a quantity increases at a consistent rate over time. This type of growth is characterized by a function that doubles, triples, or in some cases, increases by a higher factor at each step. With the function \( g(x) = 8^x \), for each increase in \( x \) by 1, the value of \( g(x) \) multiplies by 8. Here are some features of exponential growth:
  • It's non-linear, meaning it doesn鈥檛 increase by the same amount each step.
  • The rate of growth is proportional to its current value, making it accelerate as values get larger.
  • The base of the exponential function (in this case, 8) indicates the rate of this growth.
Exponential functions are used often in representing population growth, radioactive decay, and interest calculations. The visual representation of exponential growth shows a curve that initially rises slowly and then skyrockets.
Table of Values
A table of values is a fundamental tool used to determine the behavior of a function, particularly useful in graphing. In our example of the function \( g(x) = 8^x \), we use a table of values to calculate output (\( g(x) \)) based on a range of input values (\( x \)). Here鈥檚 why a table of values is crucial:
  • It allows us to see directly how the output values change with varying input values.
  • It helps in highlighting specific points that can then be plotted on a graph.
For our function:
  • \(-2\) produces \(0.0156\)
  • \(-1\) produces \(0.125\)
  • \(0\) produces \(1\)
  • \(1\) produces \(8\)
  • \(2\) produces \(64\)
Creating this table not only aids in plotting the graph but also better illustrates the nature of exponential growth by showing a rapid increase in values.
Plotting Points
Plotting points is an essential step in graphing a function, assisting in visualizing how the function's values change. For the exponential function \( g(x) = 8^x \), plotting involves drawing the points calculated from the table of values onto a graph. Here's how you can plot points effectively:
  • Each point corresponds to an \( (x, y) \) pair, where \( y = g(x) \).
  • Start by locating and marking each point on the graph paper or grid.
  • Ensure a consistent scale on both the \( x \)-axis and \( y \)-axis for accuracy.
For instance, the points \((-2, 0.0156), (-1, 0.125), (0, 1), (1, 8), (2, 64)\) should be plotted carefully. Once plotted, you can see the nature of exponential growth, especially as points move higher off the x-axis when \( x \) becomes positive, reflecting the steep rise in values.
Exponential Functions
Exponential functions form the backbone of many mathematical applications because they encapsulate rapid changes in value effectively. An exponential function such as \( g(x) = 8^x \) expresses \( g(x) \) as a power of 8, meaning the base number 8 is raised to the power of the variable \( x \). Key characteristics of exponential functions include:
  • The base of the exponentiate 鈥� a larger base indicates faster growth.
  • As \( x \) tends to negative infinity, the value of the function approaches zero, yet never exactly reaches it.
  • As \( x \) becomes large and positive, the function's values increase sharply.
  • Typically, the graph of an exponential function is a curve that rises or falls exponentially.
These functions are often involved in calculating compound interest, population models, decay processes, and other phenomena where quantities change rapidly. Understanding how to handle these functions, graph them, and interpret their tables of values is essential in many fields of science and mathematics.

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Most popular questions from this chapter

\(29-43\) . These exercises deal with logarithmic scales. Ion Concentration The pH reading of a sample of each substance is given. Calculate the hydrogen ion concentration of the substance. (a) Vinegar: pH \(=3.0\) (b) Milk: \(\mathrm{pH}=6.5\)

Draw the graph of the function in a suitable viewing rec- tangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=\ln \left(x^{2}-x\right) $$

Draw the graph of the function in a suitable viewing rec- tangle, and use it to find the domain, the asymptotes, and the local maximum and minimum values. $$ y=\frac{\ln x}{x} $$

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Atmospheric pressure \(P\) (in kilo-pascals, kPa) at altitude \(h\) (in kilometers, km) is governed by the formula $$\ln \left(\frac{P}{P_{0}}\right)=-\frac{h}{k}$$ where \(k=7\) and \(P_{0}=100 \mathrm{kPa}\) are constants. (a) Solve the equation for \(P .\) (b) Use part (a) to find the pressure \(P\) at an altitude of 4 \(\mathrm{km}\) .
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