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Algebra 1

Unit 12: Lesson 1

Exponential vs. linear growth
Math
Algebra 1
Exponential growth & decay
Exponential vs. linear growth
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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:
x12151821
yminus, 251219
Here, the x-values increase by exactly 3 units each time,
x\curvearrowright, plus, 3\curvearrowright, plus, 3\curvearrowright, plus, 3
12151821
and the y-values increase by a constant difference of 7.
y\curvearrowright, plus, 7\curvearrowright, plus, 7\curvearrowright, plus, 7
minus, 251219
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:
x0123
y13927
Here, the x-values increase by exactly 1 unit each time,
x\curvearrowright, plus, 1\curvearrowright, plus, 1\curvearrowright, plus, 1
0123
and the y-values increase by a constant factor of 3.
y\curvearrowright, times, 3\curvearrowright, times, 3\curvearrowright, times, 3
13927
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:
x2468
y491625
Here, the x-values increase by exactly 2 units each time.
x\curvearrowright, plus, 2\curvearrowright, plus, 2\curvearrowright, plus, 2
2468
However, the differences between the y-values aren't constant,
y\curvearrowright, plus, 5\curvearrowright, plus, 7\curvearrowright, plus, 9
491625
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
491625
Therefore, this relationship is neither linear nor exponential.

Check your understanding

Problem 1
x0123
y5101520
Fill in the blanks.
This relationship is
because each y-value is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
the value before it.

Problem 2
x0123
y261854
Fill in the blanks.
This relationship is
because each y-value is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
the value before it.

Problem 3
Coordinate grid with points marked at (0, 1); at (1, 2); at (2, 4); and at (3, 8).
Fill in the blanks.
This relationship is
because each y-value is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
the value before it.

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