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## Algebra 1

### Course: Algebra 1>Unit 12

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.