Problem 24 - College Algebra with Corequisite Support

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. $$ \begin{array}{ccccc} x & 1 & 2 & 3 & 4 \\ h(x) & 70 & 49 & 34.3 & 24.01 \end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Function

In this step, we determine if the given table represents a linear, exponential, or neither function by examining the values of \( h(x) \). For it to be linear, \( h(x) \) must increase or decrease by the same amount. For it to be exponential, it must increase or decrease by a consistent multiplicative factor.

Step 2: Check for Linearity

Calculate the difference between consecutive \( h(x) \) values. We find: \( 49 - 70 = -21 \), \( 34.3 - 49 = -14.7 \), \( 24.01 - 34.3 = -10.29 \). Since these differences are not constant, the function is not linear.

Step 3: Check for Exponential Growth or Decay

Calculate the ratio of consecutive \( h(x) \) terms to check for consistency:\( \frac{49}{70} = 0.7 \), \( \frac{34.3}{49} \approx 0.7 \), \( \frac{24.01}{34.3} \approx 0.7 \). Since the ratios are approximately the same, the function is likely exponential.

Step 4: Determine Exponential Function

An exponential function takes the form \( h(x) = ab^x \). We know \( h(1) = 70 \) and that \( b \approx 0.7 \). Using \( 70 = ab^1 \), we have \( a \times 0.7 = 70 \), which gives \( a = 100 \). Thus, \( h(x) = 100 imes 0.7^x \).

Key Concepts

Linear Functions

Linear functions are mathematical expressions where the relationship between the input variable and the output variable is a straight line when graphed. In simpler terms, the change between the output values is consistent each time the input increases or decreases. This change is often referred to as the slope of the line.

Key characteristics of linear functions include:

• The graph of a linear function is a straight line.
• The function can be written in the form \( f(x) = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• The change in the function's output is constant for each unit of change in the input value, meaning the difference between consecutive terms remains the same.

To determine if data is linear, check the first differences of consecutive output values. If it's constant, like in the pattern from a sequence, the function is linear. In our exercise, differences like \( -21 \), \( -14.7 \), and \( -10.29 \) showed variability, indicating the function isn't linear.

Function Analysis

Function analysis involves investigating the relationship between inputs and outputs of a function to classify it as linear, exponential, or neither. The process uses evaluations of differences and ratios to find consistency in changes—either additive or multiplicative.

Steps to analyze a function:

• Calculate the differences between outputs to assess linearity.
• Calculate the ratios of consecutive outputs to evaluate exponential behavior.
• Determine if neither pattern exists by examining variability in differences or ratios.

In our example, differences in outputs weren't consistent, ruling out linearity. However, the ratios of \( h(x) \) values approximated a consistent 0.7, suggesting exponential behavior. Thus, function analysis guided us toward identifying the function's exponential nature.

Sequence Patterns

Sequence patterns are essential in identifying the kind of function that best describes a given set of data. A sequence is a list of numbers that follow a certain rule, which can be expressed linearly or exponentially.

Linear sequences show a constant difference between terms, forming an arithmetic sequence, while exponential sequences have a constant ratio between terms, forming a geometric sequence.

• Arithmetic Sequence: The difference between consecutive terms is constant. Example: 2, 4, 6, 8 (common difference of 2).
• Geometric Sequence: The ratio between consecutive terms is constant. Example: 3, 6, 12, 24 (common ratio of 2).

In the exercise, examining the sequence of \( h(x) \) values, we see that while differences fluctuate, the ratios are almost consistent, indicating a geometric pattern. Recognizing this pattern, we conclude the sequence aligns with the characteristics of an exponential sequence, which helps in deducing the appropriate function form.