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## SAT

### Course: SAT>Unit 6

Lesson 5: Problem Solving and Data Analysis: lessons by skill

# Linear and exponential growth | Lesson

## What are linear and exponential growth problems, and how frequently do they appear on the test?

Consider the following: Alphonse and Bekah both have 100 followers on a social media platform. Over the next 12 months:
• Alphonse's number of followers increases by 10 each month.
• Bekah's number of followers increases by 10, percent each month.
Linear and exponential growth problems are all about understanding and comparing scenarios like the ones above.
In this lesson, we'll learn to:
1. Determine whether two variables have a linear or exponential relationship based on their values
2. Determine whether a real-world scenario exhibits linear or exponential growth
On your official SAT, you'll likely see 1 question that tests your ability to distinguish between linear and exponential growth.
This lesson builds upon the following skills:
You can learn anything. Let's do this!

## How do I choose between linear and exponential equations when modeling a table of values?

### Exponential vs. linear growth

Exponential vs. linear growthSee video transcript

### Writing equations based on tables

When we're given a table of left parenthesis, x, comma, y, right parenthesis values, for a given change in x:
• If the change in y can be represented by repeatedly adding the same value, then the relationship is best modeled by a linear equation.
• If the change in y can be represented by repeatedly multiplying by the same value, then the relationship is best modeled by an exponential equation.
Once we determine the correct type of equation to use, we can write the equation by using our knowledge of linear and exponential equations.
Using y, equals, m, x, plus, b to represent a linear equation:
• m is the number repeatedly added, the rate of change, or the slope of the line when the equation is graphed in the x, y-plane.
• b is the initial value, or the y-intercept of the line when the equation is graphed in the x, y-plane.
Using y, equals, a, left parenthesis, b, right parenthesis, start superscript, x, end superscript to represent an exponential equation:
• b is the number repeatedly multiplied, or the common factor or common ratio.
• a is the initial value, or the y-intercept of the curve when the equation is graphed in the x, y-plane.

Let's look at some examples!
x01234
y357911

x01234
y36122448

### Try it!

Try: write an equation based on a table of values
xy
01
18
215
322
429
In the table above, as x increases by 1, y
. Therefore, the relationship between y and x is best modeled with
.
Write an equation for y in terms of x.
y, equals

## How do I choose between linear and exponential functions to model real-world scenarios?

### Exponential vs. linear models: verbal

Exponential vs. linear models: verbalSee video transcript

### What are some common phrases to look out for?

On the SAT, all linear and exponential growth questions are multiple choice. We'll be asked to:
• Choose the correct description of a scenario from two linear and two exponential descriptions
• Choose the correct modeling equation for a scenario from two linear and two exponential equations
This means as soon as we figure out whether a relationship is linear or exponential, we can immediately eliminate two of the four choices!
After that:
• If the choices are descriptions, we need to figure out whether the relationship is increasing or decreasing.
• If the choices are models, we need to use our knowledge of linear and exponential word problems to find the equation that includes the right values.
The table below lists some common phrases in linear and exponential growth problems and how to interpret them.
Note: c is a constant in the phrases.
PhraseLinear or exponential relationship?
Changes (i.e., increases or decreases) at a constant rateLinear
Changes by c per unit of timeLinear
Changes by c, percent (of the current value) per unit of timeExponential ("Of the current value" is often implied.)
Changes by c, percent of the initial value per unit of timeLinear (Since the initial value is constant, a percent of the initial value is also constant.)
Changes by a factor of c (e.g., halves, doubles) per unit of timeExponential

### Try it!

try: match scenarios to their descriptions
Match each of the four scenarios below to their appropriate description.

Practice: write an equation based on a table of values
x12345
y510204080
Which of the following equations relates y to x for the values in the table above?

Practice: describe a modeling function
Norman was 52 centimeters long when he was born. Over the next 6 months, his length grew at a rate of 2, point, 5 centimeters a month. If f, left parenthesis, t, right parenthesis is Norman's length in centimeters t months after his birth, which of the following statements best describes the function f ?

Practice: write an equation that models a scenario
A bakery is giving away 600 cookies. The giveaway starts on a busy weekend, and passersby take the free cookies at a constant rate. After 2 hours, the bakery has given away 50, percent of the cookies. Which of the following equations models the number of cookies, C, remaining h hours after the giveaway starts?

## Want to join the conversation?

• I don't really get how the y-intercept of that one problem is 5/2, and the explain didn't really help, could you elaborate more.
• The y value is 5 when x=1, but in order to find the initial value which is the y intercept, x needs to be 0 because a y intercept is at (0,y). You can infer that if y doubles when x increases by 1, y will half when x decreases by 1. 5/2 is the same as 5*1/2 if that helps.
• Why do you subtract 150?
• They first find 50% of 600. Then they divide 300 by 2 which is 50% of 600 because the equation says that the cookies were distributed in 2 hours. Hence you have 150 cookies subtracted from 600 per hour. And that's you answer choice A. I hope it helped!
• Can't an exponential function increase at a constant rate? Is it necessary that if the question mentions "increases or decreases at a constant rate" then that function is linear?
• A linear function increases by a constant value, and an exponential function increases by a constant fraction of the previous value. If the "rate" that sentence is talking about is the ratio between the change in y to change in x (slope), then if it's constant the function is linear. If the "rate" means that the ratio between subsequent values is constant, then the function would be exponential.
So technically, having a constant rate could mean either having a constant rate of change or common ratio, and so it could mean either exponential or linear. I think it's more used for linear functions, though. Just remember to choose an exponential equation if you see something like half-lives, bacterial growth, compound interest, etc.
• how many hours should I study a day if I want to get 1500 sat score in 2 months ?
• A bakery is giving away 500 cookies. The giveaway starts on a busy weekend, and passersby take the free cookies at a constant rate. After 2 hours, the bakery has given away 50% of the cookies. Which of the following equations models the number of cookies, C, remaining h hours after the giveaway starts?
(1 vote)
• In the problem, you look for a couple things. One is that the bakery starts with 500 cookies. Another is that it gives these out at a constant rate. And the third is that after 2 hours 50% of the cookies have been given away (or 250 are remaining). With this information, we can create an equation that models this:

Since the bakery gives away cookies at a constant rate, we can use a linear equation. These take the form y = mx + b. We know b because it is our point where x is 0, or before any hours have passed and the bakery has all 500 cookies. To get the slope, we can use rise over run or plug in an (x,y) pair into the equation that we have so far in order to find m. Both are about equally as fast:
y = mx + 500
250 = m(2) + 500
-250 = 2m
m = -125
So now we know that the number of cookies remaining decreases by 125 cookies per hour. That's all we need to create our equation, just the slope and y-intercept:
y = -125x + 500