# Linear and exponential growth on the SAT test

### SAT Subscore: Problem solving and data analysis

Before learning linear growth topic you should master the linear functions topic; before learning exponential growth topic you should master the graphing exponential functions topic.

**Linear growth **occurs then a variable is growing by the same amount in each unit of time. The linear growth formula is y=mx+b, where b is the initial value and m is the constant rate of change.

**Exponential growth **occurs when the variable is growing by the same __relative__ amount in each unit of time. The exponential growth formula is y=a(b)^{x}, where a is the initial value, b is the number that is multiplied and x is the number of the multiplications.

**The difference between linear and exponential growth:** Linear growth is always at the same rate, while the rate of the exponential growth increases over time.

**Model a table with a linear equation** if the value in the table changes by a constant amount.

**Model a table with an exponential equation** if the value in the table changes by a constant multiplication factor (the common factor).

**Modeling a scenario with linear and exponential functions** questions require determining the type of the connection (linear or exponential) and the direction of the connection (increasing or decreasing).

- If the change is at a constant rate (by a constant), the relationship is linear.
- If the change is by a percent from the initial value, the relationship is linear.
- If the change is by a factor, the relationship is exponential.
- If the change is by a percent (from the previous value), the relationship is exponential.

__Continue reading this page for detailed explanations and examples.__

## Linear and exponential growth definitions

**Linear growth** (decrease) occurs then a variable is growing (decreasing) by the same amount in each unit of time, meaning that the variable gets larger (smaller) by a constant amount in each time unit. For example: the money in the saving account increases (decreases) by 1,000 dollars every month.

**The linear growth formula** is y=mx+b, where b is the initial value (the intercept) and m is the constant rate of change (the slope).

**linear growth function:** If the variable is growing, the function is increasing, meaning that the slope of the line (m) is positive and the line moves up from left to right. If the variable is declining, the function is decreasing, meaning that the slope of the line (m) is negative and the line moves down from left to right. If m=0 the variable stays constant and the line is flat.

**Exponential growth** (decay) occurs when the variable is growing (decreasing) by the same __relative__ amount in each unit of time, meaning that the growth (decay) rate gets faster in each time unit. For example: The money in the deposit accrues an interest of 5 percent per year and is deposited for 5 years.

**The exponential growth formula** is y=a(b)^{x}, where a is the initial value (the intercept), b is the number that is multiplied (the common factor or the common ratio) and x is the number of the multiplications (in time questions x is the number of periods).

**Exponential growth** **function**: In exponential growth, the function goes up from left to right; in exponential decay, the function goes down from left to right. The slope of the function gets bigger (in absolute value) as we move from left to right (the graph gets steeper).

**The difference between linear and exponential growth:** Linear growth is always at the same rate, while the rate of the exponential growth increases over time. Meaning that in exponential growth as the variable gets larger, the rate of its growth also gets larger.

## Modeling tables with linear and exponential functions

In these questions, we need to find the function that represents a given table.

#### Modeling a table with a linear function

**Model a table with a linear equation** if the value in the table changes by a constant amount.

**The linear function formula** is y=mx+b, where b is the initial value (the intercept) and m is the constant rate of change (the slope).

**Modeling a table with a linear function steps:****Step 1: **Determine if the function that represents the table is linear or exponential:

To find the correct function type check if __the change in the y variable__ is linear or exponential. If the change in the y variable is linear then we need to find a linear function equation that represents it. Note that the change in the x variable must be linear.

**Step 2: **Find the function equation using the formula y=mx+b for the linear function:

To find the intercept b, look what is the value of y when x=0 in the table. If the y value for x=0 is not given, calculate it using the change in the y variable that you found in step 1.

To find the slope m, divide the change in variable y by the change in variable x.

**Step 3: **Check the equation from step 2 by plugging into it the x values from the table. You must get y values like in the table.

**Step 4:** If needed, draw the graph the function according to its equation.

Consider the following example:

Write an equation based of the following table and find its graph.

x 0 2 4 6 8

y 3 7 11 15 19

**Step 1:** **Determining if the function that represents the table is linear or exponential:**

We need to make sure that the change in the x variable is linear; the x variable values are 0, 2, 4, 8 and 8. Their changes are constant and are equal to 2 (2-0=2, 4-2=2…).

We need to check the change in the y variable; the y variable values are 3, 7, 11, 15 and 19. Their changes are constant and are equal to 4 (7-3=4, 11-7=4, 15-11=4, 19-15=4).

**Step 2:** **Finding the function equation using the formula y=mx+b:**

For the intercept b, we look what is the value of y when x=0 in the table. If x=0 then y=3, therefore b=3.

Note that if the y value for x=0 was not given, we can calculate it using the change in the y variable that we found in step 1. The change in the y variable was equal to 4 and we know from the table that for x=2 y=7. Therefore, to find what is the y for x=0 we need to subtract 4 from 7 getting 3. If the first y value is 3 then the second y value is 3+4=7.

For the slope m, we divide the change in variable y by the change in variable x. We know from step 1 that the change in the x variable is 2 and the change in the y variable is 4. Therefore, the slope of the function is m=4/2=2.

The equation of the linear function that represents the given table is y=mx+b, y=2x+3.

**Step 3: Checking the equation:**

y=2x+3

y(x=0)=2*0+3=3

y(x=2)=2*2+3=7

**Step 4:** **Drawing the graph** the function according to its equation.

#### Modeling a table with an exponential function

**Model a table with an exponential equation** if the value in the table changes by a constant multiplication factor (the common factor).

**The exponential function formula** is y=a(b)^{x}, where a is the initial value (the intercept), b is the number that is multiplied (the common factor) and x is the number of the multiplications.

**Modeling a table with an exponential function steps:**

**Step 1:** Determine if the function that represents the table is linear or exponential- To find the correct function type check if __the change in the y variable__ is linear or exponential. If the change in the y variable is exponential, then we need to find an exponential function equation that represents it. Note that the change in the x variable must be linear.

**Step 2:** Find the function equation using the formula y=a(b)^{x} for the exponential function.

To find the initial value a, look what is the value of y when x=0 in the table. If the y value for x=0 is not given, calculate it using the change in the y variable that you found in step 1.

The common factor b is equal to the change in the variable y.

**Step 3:** Check the equation from step 2 by plugging into it the x values from the table. You must get y values like in the table.

**Step 4:** If needed, draw the graph the function according to its equation.

Consider the following example:

Write an equation based of the following table and find its graph.

x 0 1 2 3

y 2 6 18 54

**Step 1: Determining if the function that represents the table is linear or exponential:**

We need to make sure that the change in the x variable is linear; the x variable values are 0, 1, 2 and 3. Their changes are constant and are equal to 1 (1-0=1, 2-1=1…).

We need to check the change in the y variable; the y variable values are 2, 6, 18 and 54. Their changes are exponential by a factor of 3 (6:2=3, 18:6=3, 54-18=3).

**Step 2:** **Finding the function equation using the formula y=a(b) ^{x}:**

For the initial value a, we look what is the value of y when x=0 in the table. If x=0 then y=2, therefore a=2.

Note that if the y value for x=0 was not given, we can calculate it using the change in the y variable that we found in step 1. The change in the y variable was by a factor of 3 and we know from the table that for x=1 y=6. Therefore to find what is the y for x=0 we need to divide 6 by 3 getting 2. If the first y value is 2 then the second y value is 2*3=6.

The common factor b is the number that is multiplied, it was calculated in step 1 as the change in the y variable and is equal to 3.

The equation of the exponential function that represents the given table is y=a(b)^{x}, y=2(3)^{x}.

**Step 3:** **Checking the equation:**

y=2(3)^{x}

y(x=0)=2(3)^{0}=2*1=2

y(x=1)=2(3)^{x}=2(3)^{1}=2*3=6

y(x=2)=2(3)^{x}=2(3)^{2}=2*3*3=18

y(x=3)=2(3)^{x}=2(3)^{3}=2*3*3*3=54

**Step 4:** **Drawing the graph** the function according to its equation.

## Modeling a scenario (description) with linear and exponential functions

In these questions, we receive a verbal scenario and need to determine the type of the connection (linear or exponential) and the direction of the connection (increasing or decreasing) from a multiple choice equations.

#### Determining if the function that represents the scenario is linear or exponential

**If the change is at a constant rate (by a constant), the relationship is linear. **

For example: The value of the account each month is 100 dollars bigger than the value in the previous month. If the initial value is 1,000 dollars, after 1 month it will be 1,000+100=1,100 dollars, after 2 months it will be 1,100+100=1,200 dollars etc. The function equation is y=1,000+100x.

**If the change is by a percent from the initial value, the relationship is linear. **

For example: The value of the account increases by 5 percent from the initial value each month. If the initial value is 1,000 dollars, the value of the account each month will increase by 50 dollars. After 1 month it will be 1,000+50=1,050 dollars, after 2 months it will be 1,050+50=1,100 dollars etc. The function equation is y=1,000+50x.

**If the change is by a factor, the relationship is exponential. **

For example: The value of the account each month is 2 times bigger than the value in the previous month. If the initial value is 1,000 dollars, after 1 month it will be 1,000*2=2,000 dollars, after 2 months it will be 1,000*2*2=4,000 dollars etc. The function equation is y=1,000*2^{x}.

**If the change is by a percent (from the previous value), the relationship is exponential. **

For example: The value of the account increases by 5 percent (from the value in the previous month). If the initial value is 1,000 dollars, after 1 month it will be 1,000*1.05=1,050 dollars, after 2 months it will be 1,000*1.05*1.05=1,102.5 dollars etc. The function equation is y=1,000*1.05^{x}.

#### Determining if the function that represents the scenario is increasing or decreasing

All the examples above were for an increasing connection. For **decreasing** connections, the functions will become:

If the value of the account each month is 100 dollars smaller than the value in the previous month, the function y=1,000+100x will become y=1,000-100x.

If the value of the account decreases by 5 percent from the initial value each month, the function y=1,000+50x will become y=1,000-50x.

If the value of the account each month is 2 times smaller than the value in the previous month, the function y=1,000*2^{x} will become y=1,000*(1/2)^{x}.

If the value of the account decreases by 5 percent (from the value in the previous month), the function y=1,000*1.05^{x} will become y=1,000*(1-0.05)^{x}.