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Chapter 4: Problem 4

For each table below, could the table represent a function that is linear, exponential, or neither? $$ \begin{array}{|c|l|l|l|l|} \hline \mathbf{x} & 1 & 2 & 3 & 4 \\ \hline \mathbf{h}(\mathbf{x}) & 70 & 49 & 34.3 & 24.01 \\ \hline \end{array} $$

Short Answer

Expert verified
The table represents an exponential function.

Step by step solution

01

Check for Linear Function

A linear function has the form \( h(x) = mx + c \) where \( m \) is the constant rate of change (slope). To determine if the function is linear, calculate the differences between consecutive \( h(x) \) values:\[\begin{align*}\Delta h(1,2) & = h(2) - h(1) = 49 - 70 = -21 \\Delta h(2,3) & = h(3) - h(2) = 34.3 - 49 = -14.7 \\Delta h(3,4) & = h(4) - h(3) = 24.01 - 34.3 = -10.29\end{align*}\]The differences are not constant, so the function is not linear.
02

Check for Exponential Function

An exponential function has the form \( h(x) = a \cdot b^x \) where \( b \) is the constant ratio. To determine if the function is exponential, calculate the ratios of consecutive \( h(x) \) values:\[\begin{align*}h(x=2)/h(x=1) & = 49 / 70 = 0.7 \h(x=3)/h(x=2) & = 34.3 / 49 \approx 0.7 \h(x=4)/h(x=3) & = 24.01 / 34.3 \approx 0.7\end{align*}\]The ratios are approximately constant, indicating the function can be exponential.

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Key Concepts

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

Linear Functions
Linear functions are one of the simplest types of functions in mathematics. They can be easily identified by their straight-line graph and are represented by the equation \( h(x) = mx + c \). Here, \( m \) is the slope, or the rate of change, and \( c \) is the y-intercept, the point where the graph crosses the y-axis.
  • **Straight Line:** The graph of a linear function is a straight line. This means that the function has a constant rate of change.
  • **Slope \( m \):** The slope of the line is the constant rate at which \( h(x) \) changes with respect to \( x \).
In our exercise, the function isn't linear because the differences between consecutive \( h(x) \) values weren't constant. Such variability in differences indicates a lack of a fixed rate of change that's necessary for a linear function.
Exponential Functions
Exponential functions can look quite different from linear functions. These types of functions are characterized by constant ratios rather than constant differences.
- **Form**: An exponential function is expressed in the form \( h(x) = a \cdot b^x \).- **Constant Ratio:** Unlike linear functions, exponential functions change by a constant ratio, which is termed as the base \( b \).
In our example, we determined that the table represented an exponential function. Each consecutive \( h(x) \) value, when divided, yields approximately the same ratio of 0.7. This consistent ratio suggests the table represents an exponential function, even if there's a slight approximation.
Table Interpretation
Understanding tables in mathematics is a critical skill, especially when looking to determine the type of function they represent.
  • **Row and Column Analysis:** In a table, rows and columns organize data points, typically with \( x \) values in one row and \( h(x) \) in another.
  • **Sequential Comparisons:** To infer patterns, compare values sequentially to see if they exhibit constant changes or ratios.
In this exercise, interpreting the table involved carefully examining the changes and ratios between consecutive \( h(x) \) values. By doing so, we determined whether these values suggest linearity or exponential behavior.
Rate of Change
The rate of change is a vital concept in mathematical functions. It helps us comprehend how a function's value changes over intervals.
- **Linear Rate of Change:** For linear functions, this is constant and described by the slope \( m \).- **Exponential Rate of Change:** For exponential functions, the rate isn't constant in the traditional sense. Instead, successive values have a constant ratio.
In our exercise, the varying differences between consecutive \( h(x) \) values showed that the rate of change was not constant in the additive sense, ruling out linearity. However, the constant multiplicative rate confirmed the function was exponential.

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