Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, 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. $$ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 1.25 & 5.75 \\ \hline 2.25 & 8.75 \\ \hline 3.56 & 12.68 \\ \hline 4.2 & 14.6 \\ \hline 5.65 & 18.95 \\ \hline 6.75 & 22.25 \\ \hline 7.25 & 23.75 \\ \hline 8.6 & 27.8 \\ \hline 9.25 & 29.75 \\ \hline 10.5 & 33.5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The scatter plot suggests a linear function.

Step 1: Enter Data into Calculator

Begin by turning on your graphing calculator and entering the given data into two lists. Enter the \( x \)-values in List 1 (L1) and the \( f(x) \)-values in List 2 (L2).

Step 2: Create Scatter Plot

Access the 'Stat Plot' feature on your graphing calculator. Turn on Plot1 and select the scatter plot type. Set the Xlist as L1 and Ylist as L2. Enter the graph view to display the scatter plot.

Step 3: Analyze Data for Linear Pattern

Look at the scatter plot to determine if the points form a straight line. If the points approximate a straight line, the data might represent a linear function.

Step 4: Analyze Data for Exponential Pattern

Check if the points in the scatter plot curve upwards, exhibiting an increasing rate as \( x \) increases. Such a pattern would suggest the data fits an exponential function.

Step 5: Analyze Data for Logarithmic Pattern

Observe if the points form a log-like curve, where increases in \( f(x) \) decrease as \( x \) gets larger. If so, the data might follow a logarithmic function.

Step 6: Compare Observations

Compare all pattern observations: linearity, exponential growth, and logarithmic behavior. Determine which fits the data best.

Key Concepts

Linear Functions

Linear functions represent relationships where changes in one variable are proportional to changes in another. This means that when you plot a linear function on a graph, you should see the data points form a straight line or something very close to it.

Characteristics of a linear function include:

• A constant rate of change or slope.
• An equation of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• Output values ( f(x) ) that change by a consistent amount as the input values ( x ) change.

For example, if your data points were exactly on a line like \( f(x) = 3x + 2 \), increasing \( x \) by 1 unit would always increase \( f(x) \) by 3 units. When analyzing scatter plots, a linear pattern is often the first to check, because it is the simplest non-constant pattern.

Exponential Functions

Exponential functions have outputs that grow or decay at a rate proportional to their current value. On a graph, they show a curve rather than a straight line. The curve rises or falls rapidly depending on whether the function is modeling growth or decay.

Properties of exponential functions include:

• An equation in the form \( y = a \, b^x \), where \( a \) is a constant that determines the starting value and \( b \) is a positive number that dictates the growth ( b > 1 ) or decay ( b < 1 ) factor.
• A rate of change that increases or decreases multiplicatively as \( x \) changes.

When you think about exponential functions graphically:

• A growth function will show a curve that becomes steeper as \( x \) increases.
• A decay function will level off and approach zero as \( x \) increases.

In the context of the scatter plot exercise, if data points show rapid increases, this might suggest an exponential relationship.

Logarithmic Functions

Logarithmic functions often represent data where the rate of increase slows down as the input continues to grow. On a graph, they appear as a curve that starts steep and gradually becomes less steep.

The essential characteristics of logarithmic functions include:

• An equation of the form \( y = a \, \log_b(x) + c \), where \( a \) scales the function, \( b \) is the logarithmic base, and \( c \) vertically shifts the graph.
• A pattern where \( f(x) \) changes quickly at first and then plateau, as \( x \) values get larger.

Logarithmic functions are particularly useful when data increases quickly at low values of \( x \) but then levels off. If the scatter plot seems to indicate such a pattern, it may suggest a logarithmic relationship.

A real-world example can be seen in pH level measure which often follows a logarithmic scale.

Graphing Calculators

Graphing calculators are powerful tools students use for visualizing mathematical data and checking for patterns, such as those suggesting linear, exponential, or logarithmic trends. These devices help take the numerical data provided, plot it, and make it easier to identify the function type that best represents the data.

Common capabilities of graphing calculators used in these scenarios include:

• Storing data in lists for x-values and corresponding y-values.
• Generating a scatter plot from these lists to visually represent the data distribution.
• Offering graphing capabilities for various function types, aiding in comparison and pattern recognition.

Using a graphing calculator for exercises like these makes it straightforward for students to experiment with different mathematical models.
They can test hypotheses about whether data fits a linear, exponential, or logarithmic function by visually comparing plotted data against theoretical curves.