Problem 24 - College Algebra

Question

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{h}(\boldsymbol{x}) & 70 & 49 & 34.3 & 24.01 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table represents an exponential function: \(f(x) = 100(0.7)^x\).

Step 1: Determine Pattern Type

First, check if the table represents a linear, exponential, or neither type of function. A linear function has a constant rate of change, whereas an exponential function will have values that change by a constant multiplicative rate. Let's calculate: - Calculate the differences between the successive terms: - From 70 to 49: 49 - 70 = -21 - From 49 to 34.3: 34.3 - 49 = -14.7 - From 34.3 to 24.01: 24.01 - 34.3 = -10.29 The differences are not consistent, so it is not linear.

Step 2: Check for Exponential Pattern

Next, we check whether this is an exponential function by calculating the ratio of successive terms:- Calculate the ratios between successive terms: - From 70 to 49: \( \frac{49}{70} \approx 0.7 \) - From 49 to 34.3: \( \frac{34.3}{49} \approx 0.7 \) - From 34.3 to 24.01: \( \frac{24.01}{34.3} \approx 0.7 \)Since the rate of change is constant at approximately 0.7, the function is likely exponential.

Step 3: Find Exponential Function

Since the pattern suggests exponential behavior, the general formula of an exponential function is given by \( f(x) = ab^x \).- Use the first value, \(h(1) = 70\): \( 70 = ab^1 \rightarrow 70 = ab \)- Use the next value, \(h(2) = 49\): \( 49 = ab^2 \)- Divide the two equations to find \(b\): \( \frac{49}{70} = \frac{ab^2}{ab} \Rightarrow b \approx 0.7 \)Now substitute back to find \(a\):- \( 70 = a(0.7) \rightarrow a = 100 \)Thus, the exponential function is \(f(x) = 100(0.7)^x\).

Key Concepts

Linear Functions

Linear functions are a fundamental concept in mathematics that describe relationships with a constant rate of change. This means that, as one variable increases, the other variable increases or decreases by a fixed amount each time.

This fixed amount is often referred to as the "slope," and you can think of it as the steepness of a line when graphed on a coordinate plane. The basic form of a linear function can be expressed using the equation:

• \( f(x) = mx + b \)

here, \( m \) is the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.

When examining data tables, you determine a linear relationship by checking if the difference between consecutive y-values is constant. If the increase or decrease between each y-value is uniform, the function is considered linear. In the given exercise, the calculated differences \(-21, -14.7,\) and \(-10.29\) were not consistent, indicating that the table does not represent a linear function.

Patterns in Tables

Identifying patterns in tables is an essential step in determining the type of function represented by a set of data. In math, by visually exploring the tabulated data, you can spot potential relationships between the variables.

There are two main types of functions to examine: linear and exponential. When you see a table, you first check for linear patterns by calculating differences between consecutive y-values, as shown before.

If these differences are consistent, then the table represents a linear function. If not, you then explore the possibility of an exponential pattern by calculating the ratios of successive y-values:

• Exponential functions show a constant ratio, known as the growth or decay rate, rather than a constant difference.
• In our example, the ratios were all approximately \( 0.7 \), which is indicative of an exponential function.

Understanding patterns in tables helps us predict future values, recognize trends, and comprehends the underlying relationship between variables.

Function Determination

Function determination is a critical skill in mathematics, as it involves identifying the nature and equation of a function from data. This process allows you to mathematically describe how variables are associated.

To determine the function type, you can start with these steps:

• Calculate the difference between consecutive values for linear patterns.
• Compute the ratio of consecutive values for exponential patterns.

Once you've recognized the pattern, the next step is to derive the equation. For exponential functions, the form is:

• \( f(x) = ab^x \)

This formula includes:

• \( a \): the initial value or y-intercept when \( x = 0 \)
• \( b \): the growth or decay factor

In our case study, with a constant ratio of \( 0.7 \), we calculated the initial value \( a = 100 \), resulting in \( f(x) = 100(0.7)^x \). By understanding and determining the function, you can not only match the data but also gain insights into how systems behave over time.