Change in Linear | undefined Unit 2 Study Guide

The Core Idea: Change in Linear and Exponential Functions

This topic explores the fundamental difference in how linear and exponential functions grow or decay. The central concept is distinguishing between two types of constant change. Linear functions are characterized by a constant rate of change, meaning for any uniform change in the input, the output changes by a constant additive amount. This constant rate is also known as the slope.

In contrast, exponential functions are characterized by a constant multiplicative factor. This means that for each single-unit increase in the input variable, the output is multiplied by a constant value, known as the base of the exponential function. Understanding whether a function's output changes by adding a fixed number or multiplying by a fixed number is the key to identifying its underlying model as either linear or exponential.

Key Formulas & Rules

The way we measure and describe change differs significantly between linear and exponential functions. The following formulas are essential for this analysis.

Linear Function Rate of Change (Slope)

For any linear function $f$ , the rate of change is constant. This rate, called the slope $m$ , can be calculated from any two distinct points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on the function's graph.

$m = \frac{change in output}{change in input} = \frac{Δ y}{Δ x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

This formula reveals that for a linear function, the change in the output ( $Δ y$ ) is directly proportional to the change in the input ( $Δ x$ ).

Exponential Function Multiplicative Factor (Base)

For an exponential function of the form $f (x) = a \cdot b^{x}$ (where $b > 0$ and $b \neq = 1$ ), the function changes by a constant multiplicative factor, $b$ , for each unit increase in the input x.

The ratio of output values for any two inputs $x_{1}$ and $x_{2}$ where the inputs are separated by a constant interval $k$ (i.e., $x_{2} = x_{1} + k$ ) is given by:

$\frac{f ( x _{2} )}{f ( x _{1} )} = \frac{a \cdot b ^{x_{2}}}{a \cdot b ^{x_{1}}} = \frac{a \cdot b ^{x_{1} + k}}{a \cdot b ^{x_{1}}} = b^{k}$

This formula shows that the ratio of outputs depends only on the interval between the inputs ( $k$ ), not on the specific input values themselves. If the interval is one unit ( $k = 1$ ), the ratio is simply the base, $b$ .

Understanding Additive vs. Multiplicative Change

The defining characteristic that separates linear and exponential functions is the nature of their change. Mastering this distinction is crucial for modeling real-world phenomena.

A function exhibits additive change if a constant change in the input variable results in a constant amount being added to the output variable. This is the hallmark of a linear function. For example, if increasing $x$ by 1 always causes $y$ to increase by 5, the function is linear with a slope of 5. If increasing $x$ by 3 always causes $y$ to decrease by 6, the function is linear with a slope of $- 6/3 = - 2$ . The key operation is addition (or subtraction).

A function exhibits multiplicative change if a constant change in the input variable results in the output variable being multiplied by a constant factor. This is the hallmark of an exponential function. For example, if increasing $x$ by 1 always causes $y$ to be multiplied by 2, the function is exponential with a base of 2. If increasing $x$ by 1 causes $y$ to be multiplied by 0.5 (i.e., halved), the function is exponential with a base of 0.5. The key operation is multiplication.

This distinction is most easily observed in a table of values where the $x$ -values increase by a constant increment. To test for a linear function, you calculate the differences between consecutive $y$ -values. To test for an exponential function, you calculate the ratios of consecutive $y$ -values.

Core Concepts & Rules

Linear Function Change: A function is linear if it has a constant rate of change (slope).
Calculating Linear Change: The constant rate of change, $m$ , is found by calculating the ratio of the change in output to the change in input ( $Δ y /Δ x$ ) between any two points.
Proportionality in Linear Functions: In a linear function, the change in the output is directly proportional to the change in the input.
Exponential Function Change: A function of the form $f (x) = a \cdot b^{x}$ is exponential if it changes by a constant multiplicative factor for each unit increase in the input.
Identifying Exponential Change: This constant multiplicative factor is the base, $b$ , of the exponential function.
Ratios in Exponential Functions: For an exponential function, the ratio of outputs $f (x_{2}) / f (x_{1})$ over an interval $x_{2} - x_{1} = k$ is constant and equal to $b^{k}$ .

Step-by-Step Example 1: Analyzing Change in a Table

Problem: A function $f (x)$ is represented by the table of values below. Determine if the function is linear or exponential and find its constant rate of change or constant multiplicative factor.

$x$	$f (x)$
2	5
4	11
6	17
8	23

Step 1: Check for Linear Change (Constant Additive Change)

We calculate the rate of change ( $Δ y /Δ x$ ) between consecutive pairs of points to see if it is constant.

Between $(2, 5)$ and $(4, 11)$ :
$Δ x = 4 - 2 = 2$
$Δ y = 11 - 5 = 6$
Rate of change = $Δ y /Δ x = 6/2 = 3$
Between $(4, 11)$ and $(6, 17)$ :
$Δ x = 6 - 4 = 2$
$Δ y = 17 - 11 = 6$
Rate of change = $Δ y /Δ x = 6/2 = 3$
Between $(6, 17)$ and $(8, 23)$ :
$Δ x = 8 - 6 = 2$
$Δ y = 23 - 17 = 6$
Rate of change = $Δ y /Δ x = 6/2 = 3$

Step 2: Analyze the Results

The rate of change between every pair of consecutive points is a constant value of 3.

Step 3: Conclusion

Because the function has a constant rate of change, the function $f (x)$ is linear. Its constant rate of change (slope) is $m = 3$ .

Step-by-Step Example 2: Exam-Style Application

Problem: The table below shows some values for a function $g (x)$ . The $x$ -values are increasing by a constant interval. Determine if $g (x)$ could be modeled by a linear or an exponential function. Justify your answer by finding the constant rate of change or the constant multiplicative factor.

$x$	$g (x)$
0	80
3	40
6	20
9	10

Step 1: Test for a Linear Model

Calculate the rate of change ( $Δ y /Δ x$ ) for the first two intervals.

Interval 1 (from $x = 0$ to $x = 3$ ):
$Δ x = 3 - 0 = 3$
$Δ y = 40 - 80 = - 40$
Rate of change = $Δ y /Δ x = - 40/3 \approx - 13.33$
Interval 2 (from $x = 3$ to $x = 6$ ):
$Δ x = 6 - 3 = 3$
$Δ y = 20 - 40 = - 20$
Rate of change = $Δ y /Δ x = - 20/3 \approx - 6.67$

Since the rates of change ( $- 40/3$ and $- 20/3$ ) are not equal, the function is not linear.

Step 2: Test for an Exponential Model

Calculate the ratio of consecutive outputs, $g (x_{2}) / g (x_{1})$ . The input interval is constant: $Δ x = 3$ . According to the rule $f (x_{2}) / f (x_{1}) = b^{k}$ , we should expect a constant ratio equal to $b^{3}$ .

Ratio for the first interval:
$g (3) / g (0) = 40/80 = 0.5$
Ratio for the second interval:
$g (6) / g (3) = 20/40 = 0.5$
Ratio for the third interval:
$g (9) / g (6) = 10/20 = 0.5$

Step 3: Analyze the Results and Conclude

The ratio of consecutive outputs is constant ( $0.5$ ). This indicates that the function $g (x)$ can be modeled by an exponential function.

Step 4: Find the Constant Multiplicative Factor (Base $b$ )

The constant ratio over an interval of $Δ x = 3$ is $0.5$ . This means $b^{3} = 0.5$ . To find the constant multiplicative factor $b$ for a unit increase in $x$ , we would solve for $b$ :

$b = (0.5)^{(} 1/3) \approx 0.7937$

However, the question asks for the constant multiplicative factor associated with the change shown in the table. The constant factor for an increase of 3 in $x$ is $0.5$ .

Justification: The function $g (x)$ is exponential because for a constant change in input, $Δ x = 3$ , there is a constant ratio of outputs of $0.5$ .

Using Your Calculator

For this topic, a graphing calculator is most useful for quickly and accurately analyzing data from a table to determine the type of function. The primary tool is the list editor.

Problem: Use a calculator to verify if the data in Example 2 represents a linear or exponential function.

Data: $x = 0, 3, 6, 9$ , $g (x) = 80, 40, 20, 10$

Step 1: Enter the Data into Lists

Press STAT and select 1:Edit....
In list `L1$, enter the $x$ -values: $0, 3, 6, 9$ .
In list `L2$, enter the $g (x)$ -values: $80, 40, 20, 10$ .

Step 2: Test for a Linear Relationship (Constant Rate of Change)

Move the cursor to the header of L3.
Press 2nd then STAT to open the LIST menu.
Navigate to the OPS (Operations) menu and select 7:ΔList(.
Enter `L2$ by pressing $2 n d$ then $2$ . Close the parenthesis: $Δ L i s t (L 2)$ .
Press ÷.
Enter the change in L1: ΔList(L1) $. 7. Your formula at the top should read: $L3=ΔList(L2)/ΔList(L1)$ . Press ENTER`.
`L3$ will display $- 13.33, - 6.67, - 3.33$ . Since these values are not constant, the function is not linear.

Step 3: Test for an Exponential Relationship (Constant Ratio)

Move the cursor to the header of L4.
We want to calculate the ratio of each term in L2 to the previous term. This is slightly more complex to automate. A straightforward method is to calculate it on the home screen or manually define the list.
Let's define L4 manually. In the L4 header, type the formula to get the ratio of L2 to a shifted version of L2. A simpler, more direct method for a short list is:
- Go to the home screen (2nd > MODE).
- Calculate $L 2 (2) / L 2 (1)$ by typing $40/80$ . Result: $0.5$ .
- Calculate $L 2 (3) / L 2 (2)$ by typing $20/40$ . Result: $0.5$ .
- Calculate $L 2 (4) / L 2 (3)$ by typing $10/20$ . Result: $0.5$ .
Since the ratio between consecutive terms is a constant $0.5$ , the data represents an exponential function.

AP Exam Quick Hit

Common Question Types

Table Identification: You will be given a table of values and asked to determine if the function represented is linear, exponential, or neither. You must justify your answer by showing that the rate of change ( $Δ y /Δ x$ ) or the ratio of consecutive outputs is constant.
- Example: "The values of a function $h$ are given in the table. Could $h$ be linear or exponential? Show the calculations that lead to your answer."
Finding the Constant Factor or Rate: Given that a function is known to be linear or exponential, you will be given two points and asked to find the slope ( $m$ ) or the base ( $b$ ).
- Example: "A linear function $f$ satisfies $f (5) = 10$ and $f (8) = 22$ . What is the rate of change of $f$ ?"
Interpreting Ratios and Differences: You may be asked a conceptual question about the meaning of change in a function.
- Example: "For an exponential function $g (x) = a \cdot (1.2)^{x}$ , find the value of the ratio $g (x + 4) / g (x)$ ."

Common Mistakes

Confusing Ratios and Differences: The most common error is to calculate differences ( $y_{2} - y_{1}$ ) when testing for an exponential function, or to calculate ratios ( $y_{2} / y_{1}$ ) when testing for a linear function. Remember: Linear = Additive Change (Differences), Exponential = Multiplicative Change (Ratios).
Ignoring Non-Unit Changes in $x$ : When calculating the slope, students often forget to divide by $Δ x$ , especially if $Δ x$ is not 1. The slope is $Δ y /Δ x$ , not just $Δ y$ .
Miscalculating the Exponential Base: When $x$ changes by an amount $k$ that is not 1, the ratio of outputs is $b^{k}$ , not $b$ . For instance, if $x$ increases by 2 and the output doubles, then $b^{2} = 2$ , which means $b = 2$ , not $b = 2$ .
Arithmetic Errors: Simple mistakes in subtraction or division when calculating slope or ratios can lead to an incorrect conclusion about the function type. Always double-check your calculations.

Change in Linear and Exponential Functions - AP PreCalculus Study Guide