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### Course: Integrated math 1>Unit 14

Lesson 1: Exponential vs. linear growth

# Warmup: exponential vs. linear growth

## 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$$↷×3$$↷×3$$↷×3$
$1$$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$$↷×\frac{9}{4}$$↷×\frac{16}{9}$$↷×\frac{25}{16}$
$4$$9$$16$$25$
Therefore, this relationship is neither linear nor exponential.

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.

## Want to join the conversation?

• Wow, how the heck do you guys stand this every minute of every day?
• By relishing the anticipation of it all being over.
• 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.
• 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.
• Why? This makes me wanna cry myself to sleep.
• I know it's hard.
• Im still confused how are we supposed to know what is exponential or linear?