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### Course: Digital SAT Math>Unit 3

Lesson 7: Linear and exponential growth: foundations

# Linear and exponential growth | Lesson

A guide to linear and exponential growth on the digital SAT

## What are linear and exponential growth problems?

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 $10\mathrm{%}$ 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
This lesson builds upon the following skills:
• Understanding linear relationships
• Quadratic and exponential word problems
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

Khan Academy video wrapper
Exponential vs. linear growthSee video transcript

### Writing equations based on tables

When we're given a table of $\left(x,y\right)$ 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 $y=mx+b$ 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 $xy$-plane.
• $b$ is the initial value, or the $y$-intercept of the line when the equation is graphed in the $xy$-plane.
Using $y=a\left(b{\right)}^{x}$ 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 $xy$-plane.
Let's look at some examples!
$x$$0$$1$$2$$3$$4$
$y$$3$$5$$7$$9$$11$
$x$$0$$1$$2$$3$$4$
$y$$3$$6$$12$$24$$48$

### Try it!

Try: write an equation based on a table of values
$x$$y$
$0$$1$
$1$$8$
$2$$15$
$3$$22$
$4$$29$
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 $x$.
$y=\phantom{\rule{0.167em}{0ex}}$

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

### Exponential vs. linear models: verbal

Khan Academy video wrapper
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 $c\mathrm{%}$ (of the current value) per unit of timeExponential ("Of the current value" is often implied.)
Changes by $c\mathrm{%}$ 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.

## Your turn!

Practice: write an equation based on a table of values
$x$$1$$2$$3$$4$$5$
$y$$5$$10$$20$$40$$80$
Which of the following equations relates $y$ to $x$ for the values in the table above?
Choose 1 answer:

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 $2.5$ centimeters a month. If $f\left(t\right)$ is Norman's length in centimeters $t$ months after his birth, which of the following statements best describes the function $f$ ?
Choose 1 answer:

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 $50\mathrm{%}$ of the cookies. Which of the following equations models the number of cookies, $C$, remaining $h$ hours after the giveaway starts?
Choose 1 answer:

## Want to join the conversation?

• Video of exponentials vs linear models example 1.
Where does the 1.5 comes from ?
If the weight increases by 5% each week and the initial weight is 40kg, in one week wouldn't the weight be of 42 ?

After all, 5% of 40 is 2.
(19 votes)
• its 1.05 not 1.5 ; The initial weight is 100% ,and it increases by 5% ,so the weight of the calf is now 105% , if you write that as decimal it's 1.05 *(original weight )
(59 votes)
• Will we get 1500, if we just use khan academy as our source?
(23 votes)
• practice papers from bluebook are necessary too
(11 votes)
• Reviewing this stuff again coz didn’t level up
(24 votes)
• In the second video why the weigh increases by 1.05? Shouldn't it increase by 0.05?
(6 votes)
• Let me ask you: what does a 0.05 times increase mean?

Well, my friend, it does not mean increase, it means decrease because you are multiplying a given number by 0.05.

However, if you multiply by any number greater than 1 then the product is larger than the number.

( the original value of anything is considered to be 100% or 1, if you increase 5% then you are adding it with the 100%; so, (100+5)% = 105% or 1.05)

If you still have doubt, then let's consider the following example:
Let's say the initial weight is 100 kg and it increases 5% everyday.
So, if you want to get the correct answer what the weight will be next day, you'll have to multiply it with 1.05 which gives you the result: 105 kg.

However, if you multiply it by 0.05, then you get 5, which does not indicate the weight you'll get next day; rather it just tells you how much weight has increased since the last day. So, if you do find the increase, you'll have to add it with your original value, then you'll get the final answer.

Hope it helps.
(22 votes)
• uuuuuhhhhhhhhhhhhhhhhhhhh uhhhhhhhh uhhhhhhhhhhhhhhhh uhhhhhhhhhhhhhhhhhhhhh?
(16 votes)
• that was an easy topic ask any question about it
(11 votes)
• where did 150 come from in the last question can you explain this please 😭
(5 votes)
• in last example ; why does the 150 come?
(2 votes)
• I have found that the easiest way to do this is to create equation in the form of y=mx+b. You're solving for cookies (C) that are left after h hours. So that means the y=mx+b becomes C=mh+b.
It tells you that the original amount of cookies after no time has passed is 600. So that's the y-intercept. Now the equation is C=mh+600.
It also tells you that after 2 hours, half of the cookies have been sold. (By saying 50%, they're trying to trick you into thinking it's an exponential function. But it says the cookies are sold at a constant rate, so it's linear.)
If half of the 600 cookies have been sold, that means there are 300 cookies remaining. That's a point on the line. (2, 300). 2 is the number of hours passed, h, and 300 is the number of cookies remaining, C.
You can plug that point into your equation to solve for m. 300=m(2)+600. Solving that, you get m=-150. So the equation is C=-150h+600. You switch around the order to get the answer: C=600-150h.
(12 votes)
• Hi everyone, is only khan academy and bluebook App enough to get us 1500+ ?
(5 votes)
• Yes it is
(1 vote)
• why is the phrase "change by c% of the initial value per unit time" representing a linear graph?
(3 votes)
• if initial value then linear and if current value then exponential
(2 votes)
• And I’m here again!
(4 votes)