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Chapter 6: Problem 15

Enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}{x} & \hline {4} & {5} & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} \\ \hline f(x) & {9.429} & {9.972} & {10.415} & {10.79} & {11.115} & {11.401} & {11.657} & {11.889} & {12.101} & {12.295}\\\ \hline \end{array}$$

Short Answer

Expert verified
The data could represent a logarithmic function.

Step by step solution

01

Enter Data into Calculator

Start by entering the `x` values into List 1 (L1) and the `f(x)` values into List 2 (L2) on your graphing calculator. Ensure all pairs of data are correctly entered.
02

Graph the Scatter Plot

Use the graphing calculator to plot the scatter plot of the data. Access the graphing functions menu and select 'Scatter Plot'. Choose L1 as the X-list and L2 as the Y-list. Then, display the graph.
03

Analyze the Scatter Plot

Observe the shape and pattern of the plotted points in the scatter plot. Look for a linear, upward or downward trending line, or an exponential curve that grows or decays, or a logarithmic pattern with rapid growth initially slowing down.
04

Compare with Function Types

Compare the scatter plot to typical graphs of linear, exponential, and logarithmic functions. A linear function would show a straight line, an exponential function would show a consistent rate of growth or decay, and a logarithmic function would show rapid growth initially that slows.
05

Determine the Function Type

Based on the pattern observed, determine which function type the data most closely resembles. In this case, if the data shows a steady increase that is not consistent with a linear growth nor rapidly accelerating growth typical of exponential functions, it might be logarithmic.

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Key Concepts

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

Graphing Calculator
Using a graphing calculator is an integral part of analyzing data patterns. It is a tool that helps visualize mathematical equations and data points. To start with, you need to input your data correctly. Typically, you will use List 1 (L1) for your independent variable, often `x` values, and List 2 (L2) for your dependent variable, usually `f(x)` values.

A scatter plot can be generated by selecting the 'Scatter Plot' option in the calculator's graphing functions menu. It's important to ensure each x-value is correctly paired with its corresponding f(x) value.
  • Enter each x-value into List 1 (L1).
  • Enter each corresponding f(x) value into List 2 (L2).
  • Select the scatter plot option in the calculator's menu.
  • Set L1 as the X-list and L2 as the Y-list.
  • Display the graph to observe the plotted points.
This visual representation is the foundation for determining the type of function that describes your data.
Function Determination
The key to function determination is observing the scatter plot's pattern and how data points are distributed. Mathematical functions generally fall into a few basic types: linear, exponential, and logarithmic. Each has distinctive characteristics in terms of growth and trend.

When you examine a scatter plot, certain patterns help indicate the underlying function type. You have to look for how data points align:
  • A straight-line pattern suggests a linear function.
  • A curve that shows rapid growth or decay suggests an exponential function.
  • A rapid initial growth that eventually levels out suggests a logarithmic function.
The goal is to match the scatter plot with these typical patterns to determine which function best describes your data.
Linear Function
A linear function is one of the simplest types of mathematical functions. Its graph is represented by a straight line and is defined by the equation \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.

When analyzing a scatter plot:
  • Look for a consistent, steady increase or decrease in y-values as x-values increase.
  • The points should align closely along a straight line.
  • Linear functions show an additive change, meaning their rate of change is constant.
If your data points closely follow a straight line on your scatter plot, then the data can likely be modeled by a linear function.
Exponential Function
Exponential functions show a different pattern than linear functions. They exhibit rates of growth or decay that are proportional to their current value. This type of function is generally represented by the equation \( y = a \cdot b^x \), where \( a \) is a constant multiplier and \( b \) is the base of the exponential.

To identify an exponential function in a scatter plot:
  • Look for data points that create a curve rather than a straight line.
  • Observe whether the y-values increase or decrease at an increasing rate.
  • Exponential growth will have a convex upward curve, whereas exponential decay will present a concave downward curve.
If your scatter plot shows this pattern, the data is likely best described by an exponential function.
Logarithmic Function
Logarithmic functions have a unique pattern, demonstrating rapid growth at first which then begins to slow. This function is expressed by the equation \( y = a \cdot \log(x) + b \), where \( a \) and \( b \) are constants.

In a scatter plot:
  • Look for data points that rise sharply initially and then level off as x increases.
  • The initial steep slope followed by a gradual flattening is typical of logarithmic growth.
  • This pattern is particularly distinguishable from the constant changes of linear functions and the ongoing acceleration or deceleration of exponential functions.
By recognizing this specific pattern in your scatter plot, you can conclude that a logarithmic function may be the best fit for your data.

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Most popular questions from this chapter

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. $$ \log (x)-\frac{1}{2} \log (y)+3 \log (z) $$

For the following exercises, sketch the graph of the indicated function. $$h(x)=-\frac{1}{2} \ln (x+1)-3$$

Can the power property of logarithms be derived from the power property of exponents using the equation \(b^{x}=m ?\) If not, explain why. If so, show the derivation.

Use the result from the previous exercise to graph the logistic model \(P(t)=\frac{20}{1+4 e^{-0.5 t}}\) along with its inverse on the same axis. What are the intercepts and asymptotes of each function?

For the following exercises, use this scenario: The equation \(N(t)=\frac{500}{1+49 e^{-0.7 t}}\) models the number of people in a town who have heard a rumor after \(t\) days. How many people started the rumor?
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