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## Algebra 1

### Unit 12: Lesson 1

Exponential vs. linear growth# Warmup: exponential vs. linear growth

CCSS.Math: , ,

## 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 | minus, 2 | 5 | 12 | 19 |

Here, the x-values increase by exactly 3 units each time,

x | \curvearrowright, plus, 3 | \curvearrowright, plus, 3 | \curvearrowright, plus, 3 | |||||
---|---|---|---|---|---|---|---|---|

12 | 15 | 18 | 21 |

and the y-values increase by a constant

*difference*of 7.y | \curvearrowright, plus, 7 | \curvearrowright, plus, 7 | \curvearrowright, plus, 7 | |||||
---|---|---|---|---|---|---|---|---|

minus, 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 | \curvearrowright, plus, 1 | \curvearrowright, plus, 1 | \curvearrowright, plus, 1 | |||||
---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 |

and the y-values increase by a constant

*factor*of 3.y | \curvearrowright, times, 3 | \curvearrowright, times, 3 | \curvearrowright, 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 | \curvearrowright, plus, 2 | \curvearrowright, plus, 2 | \curvearrowright, plus, 2 | |||||
---|---|---|---|---|---|---|---|---|

2 | 4 | 6 | 8 |

However, the differences between the y-values aren't constant,

y | \curvearrowright, plus, 5 | \curvearrowright, plus, 7 | \curvearrowright, plus, 9 | |||||
---|---|---|---|---|---|---|---|---|

4 | 9 | 16 | 25 |

and the ratios aren't constant either.

y | \curvearrowright, times, start fraction, 9, divided by, 4, end fraction | \curvearrowright, times, start fraction, 16, divided by, 9, end fraction | \curvearrowright, times, start fraction, 25, divided by, 16, end fraction | |||||
---|---|---|---|---|---|---|---|---|

4 | 9 | 16 | 25 |

Therefore, this relationship is

*neither linear nor exponential*.## Check your understanding

## Want to join the conversation?

- x|0,1, 2, 3

y|2,6,18,54

How is this exponential?(5 votes)- It's really easy, if you think about it, in x it increases by 1, tho in Y it doesn't increase correctly, for an explanation,2 plus 4= 2 but 6 plus 2 is not 18. so if we times so 2 times 3. 6 times 3 is 18. you got it! the value of y that changes is 3(6 votes)

- How can I understand the exponential growth from the graph?(6 votes)
- 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(6 votes)

- isn't it the same as geometric sequences?(4 votes)
- 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.(4 votes)

- It's Exponential because the number increases and it is always being multiplied by 2(3 votes)
- 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.(2 votes)

- Wow, how the heck do you guys stand this every minute of every day?(4 votes)
- is subtracting or negative considered linear(1 vote)
- subtracting the same number over and over is linear.(6 votes)

- Why? This makes me wanna cry myself to sleep.(3 votes)
- How do I get answers? It's confusing(2 votes)
- i dont know the last one(2 votes)
- Which characteristic makes a table exponential and which one makes linear?(2 votes)
- Linear is when you adding and when you multiply the exponential.(1 vote)