Determine whether each of the following tables of values...

Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an exponential function, find a formula for the function. A. $$\begin{array}{c|c} \hline x & f(x) \\ \hline 0 & 10.5 \\ 1 & 12.7 \\ 2 & 18.9 \\ 3 & 36.7 \\ \hline \end{array}$$ B. $$\begin{array}{c|l} \hline t & s(t) \\ \hline-1 & 50.2 \\ 0 & 30.12 \\ 1 & 18.072 \\ 2 & 10.8432 \\ \hline \end{array}$$ C. $$\begin{array}{c|c} \hline u & g(u) \\ \hline 0 & 27 \\ 2 & 24 \\ 4 & 21 \\ 6 & 18 \\ \hline \end{array}$$

Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an exponential function, find a formula for the function.
A. $$\begin{array}{c|c} \hline x & f(x) \\ \hline 0 & 10.5 \\ 1 & 12.7 \\ 2 & 18.9 \\ 3 & 36.7 \\ \hline \end{array}$$
B. $$\begin{array}{c|l} \hline t & s(t) \\ \hline-1 & 50.2 \\ 0 & 30.12 \\ 1 & 18.072 \\ 2 & 10.8432 \\ \hline \end{array}$$
C. $$\begin{array}{c|c} \hline u & g(u) \\ \hline 0 & 27 \\ 2 & 24 \\ 4 & 21 \\ 6 & 18 \\ \hline \end{array}$$

Key Concepts

Formula Derivation from Data

Deriving a formula from a table of values involves identifying either a constant difference (for a linear function) or a constant ratio (for an exponential function) and then using one of the known points to solve for any unknown constants in the formula. For a linear function, once the slope (m) and y-intercept (b) are determined, the function can be written in the standard form. For an exponential function, after finding the common ratio (r), using the initial value (a) helps in specifying the exact function form.

Linear Function

A linear function is defined by a constant rate of change between the input and output values, which means the change in the function’s value is the same for equal increments in the independent variable. Its general formula is expressed as f(x) = mx + b, where m represents the slope (or rate of change) and b represents the y-intercept, the value of the function when the independent variable is zero.

Exponential Function

An exponential function is characterized by a constant multiplicative rate of change, meaning that the ratio between successive outputs is constant when the input increases by equal increments. Its general formula is written as f(x) = a * r^x, where a is the initial value (the output when x is zero) and r is the common ratio, representing the base of the exponential growth or decay.

Difference and Ratio Analysis

When given a table of values, determining if the data fits a linear or exponential model involves analyzing either the differences or the ratios of successive outputs. For a linear function, the differences between consecutive output values should be constant, indicating a constant addition. For an exponential function, the ratio of successive outputs should be constant, meaning the function is multiplied by the same factor for each unit increase in the input.

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