How can I understand the exponential growth from the graph?

The exponential kinda does like a half of an u shape. Just like one side, now if you had both sides though it would be a full u shape

It's Exponential because the number increases and it is always being multiplied by 2

Yep, linear gets numbers added to it. Another good way to remember it's exponential s if the variable is in the exponents, and only in the exponents.

isn't it the same as geometric sequences?

They are related, but not the same. Remember that a sequence is not continuous, it has values only at whole numbers generally starting at 1, 2, 3, 4 so it is a bunch of dots on the graph. If you have the exponential function that is related to the sequence, it is continuous and goes from negative infinity to infinity.

x|0,1, 2, 3 y|2,6,18,54 How is this exponential?

The equation would be 2*3^x. Ex. 2*3^0=2*1=2; 2*3^1=2*3=6; 2*3^2=2*9=18. Each increase is by a power of 3.

If I am understanding correctly, then wouldn't geometric sequences and exponential functions be the same thing?

Not exactly, but they are similar in nature. It's just that geometric sequences deal with discrete variables whereas exponential functions deal with continuous ones.

is subtracting or negative considered linear

subtracting the same number over and over is linear.

Main content

Algebra 1

Course: Algebra 1 > Unit 12

Lesson 1: Exponential vs. linear growth

Warmup: exponential vs. linear growth

Google Classroom

Exponential vs. linear growth: review

Linear and exponential relationships differ in the way the

y

-values change when the

x

-values increase by a constant amount:

In a linear relationship, the $y$ ‍-values have equal differences.
In an exponential relationship, the $y$ ‍-values have equal ratios.

Let's see some examples

Example 1: Linear growth

Consider the relationship represented by this table:

$x$ ‍	$12$ ‍	$15$ ‍	$18$ ‍	$21$ ‍
$y$ ‍	$- 2$ ‍	$5$ ‍	$12$ ‍	$19$ ‍

Here, the

x

-values increase by exactly

3

units each time,

$x$ ‍		$↷ + 3$ ‍		$↷ + 3$ ‍		$↷ + 3$ ‍
	$12$ ‍		$15$ ‍		$18$ ‍		$21$ ‍

and the

y

-values increase by a constant difference of

7

$y$ ‍		$↷ + 7$ ‍		$↷ + 7$ ‍		$↷ + 7$ ‍
	$- 2$ ‍		$5$ ‍		$12$ ‍		$19$ ‍

Therefore, this relationship is linear because each

y

-value is

7

more than the value before it.

Example 2: Exponential growth

Consider the relationship represented by this table:

$x$ ‍	$0$ ‍	$1$ ‍	$2$ ‍	$3$ ‍
$y$ ‍	$1$ ‍	$3$ ‍	$9$ ‍	$27$ ‍

Here, the

x

-values increase by exactly

1

unit each time,

$x$ ‍		$↷ + 1$ ‍		$↷ + 1$ ‍		$↷ + 1$ ‍
	$0$ ‍		$1$ ‍		$2$ ‍		$3$ ‍

and the

y

-values increase by a constant factor of

3

$y$ ‍		$↷ \times 3$ ‍		$↷ \times 3$ ‍		$↷ \times 3$ ‍
	$1$ ‍		$3$ ‍		$9$ ‍		$27$ ‍

Therefore, this relationship is exponential because each

y

-value is

3

times the value before it.

Example 3: Growth that is neither linear nor exponential

It's important to remember there can be many relationships that describe growth but aren't linear or exponential.

For example, consider the relationship represented by this table:

$x$ ‍	$2$ ‍	$4$ ‍	$6$ ‍	$8$ ‍
$y$ ‍	$4$ ‍	$9$ ‍	$16$ ‍	$25$ ‍

Here, the

x

-values increase by exactly

2

units each time.

$x$ ‍		$↷ + 2$ ‍		$↷ + 2$ ‍		$↷ + 2$ ‍
	$2$ ‍		$4$ ‍		$6$ ‍		$8$ ‍

However, the differences between the

y

-values aren't constant,

$y$ ‍		$↷ + 5$ ‍		$↷ + 7$ ‍		$↷ + 9$ ‍
	$4$ ‍		$9$ ‍		$16$ ‍		$25$ ‍

and the ratios aren't constant either.

$y$ ‍		$↷ \times \frac{9}{4}$ ‍		$↷ \times \frac{16}{9}$ ‍		$↷ \times \frac{25}{16}$ ‍
	$4$ ‍		$9$ ‍		$16$ ‍		$25$ ‍

Therefore, this relationship is neither linear nor exponential.

Check your understanding

Problem 1

$x$ ‍	$0$ ‍	$1$ ‍	$2$ ‍	$3$ ‍
$y$ ‍	$5$ ‍	$10$ ‍	$15$ ‍	$20$ ‍

Fill in the blanks.

This relationship is

because each

y

-value is

the value before it.

Problem 2

$x$ ‍	$0$ ‍	$1$ ‍	$2$ ‍	$3$ ‍
$y$ ‍	$2$ ‍	$6$ ‍	$18$ ‍	$54$ ‍

Fill in the blanks.

This relationship is

because each

y

-value is

the value before it.

Problem 3

Fill in the blanks.

This relationship is

because each

y

-value is

the value before it.

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Karra Wallis
Posted 4 years ago. Direct link to Karra Wallis's post “x|0,1, 2, 3 y|2,6,18,54 H...”
x|0,1, 2, 3
y|2,6,18,54
How is this exponential?
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- Bob Sheehy
  Posted 4 years ago. Direct link to Bob Sheehy's post “The equation would be 2*3...”
  The equation would be 2*3^x. Ex. 2*3^0=2*1=2; 2*3^1=2*3=6; 2*3^2=2*9=18. Each increase is by a power of 3.
  Comment on Bob Sheehy's post “The equation would be 2*3...”
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riley9948
Posted 4 years ago. Direct link to riley9948's post “Wow, how the heck do you ...”
Wow, how the heck do you guys stand this every minute of every day?
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- Elaine
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  lol cause I love math :)
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ebell813
Posted 4 years ago. Direct link to ebell813's post “It's Exponential because ...”
It's Exponential because the number increases and it is always being multiplied by 2
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Answer
- loumast17
  Posted 4 years ago. Direct link to loumast17's post “Yep, linear gets numbers ...”
  Yep, linear gets numbers added to it. Another good way to remember it's exponential s if the variable is in the exponents, and only in the exponents.
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Mohammad Hadi
Posted 4 years ago. Direct link to Mohammad Hadi's post “How can I understand the ...”
How can I understand the exponential growth from the graph?
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Answer
- Morgan Kling
  Posted 4 years ago. Direct link to Morgan Kling's post “The exponential kinda doe...”
  The exponential kinda does like a half of an u shape. Just like one side, now if you had both sides though it would be a full u shape
  Comment on Morgan Kling's post “The exponential kinda doe...”
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Vinay Sharma
Posted 2 years ago. Direct link to Vinay Sharma's post “isn't it the same as geom...”
isn't it the same as geometric sequences?
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Answer
- David Severin
  Posted 2 years ago. Direct link to David Severin's post “They are related, but not...”
  They are related, but not the same. Remember that a sequence is not continuous, it has values only at whole numbers generally starting at 1, 2, 3, 4 so it is a bunch of dots on the graph. If you have the exponential function that is related to the sequence, it is continuous and goes from negative infinity to infinity.
  Comment on David Severin's post “They are related, but not...”
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dc4369934
Posted a year ago. Direct link to dc4369934's post “Why? This makes me wanna ...”
Why? This makes me wanna cry myself to sleep.
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- 508344
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  I know it's hard.
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AdorableDragonNamedBloomia
Posted 5 months ago. Direct link to AdorableDragonNamedBloomia's post “If I am understanding cor...”
If I am understanding correctly, then wouldn't geometric sequences and exponential functions be the same thing?
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(4 votes)
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- prismo
  Posted 5 months ago. Direct link to prismo's post “Not exactly, but they are...”
  Not exactly, but they are similar in nature. It's just that geometric sequences deal with discrete variables whereas exponential functions deal with continuous ones.
  Comment on prismo's post “Not exactly, but they are...”
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eytan.kokorowski
Posted 4 months ago. Direct link to eytan.kokorowski's post “How can you tell the diff...”
How can you tell the difference between exponential and quadratic?
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02-26 Tadic, Stevana
Posted 4 years ago. Direct link to 02-26 Tadic, Stevana's post “is subtracting or negativ...”
is subtracting or negative considered linear
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- David Severin
  Posted 4 years ago. Direct link to David Severin's post “subtracting the same numb...”
  subtracting the same number over and over is linear.
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  (7 votes)
Morgan717
Posted 4 years ago. Direct link to Morgan717's post “How do I get answers? It'...”
How do I get answers? It's confusing
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