LINEAR VS EXPONENTIAL GROWTH

Linear Growth

Consider the relationship represented by the table shown below.

In the table above, a constant change of +1 in x corresponds to a constant change +2 in y.

Therefore, the relationship given in the table above represents linear growth, because each y-value is 2 more than the value before it.

The points from this table lie on a line.

Exponential Growth

Consider the relationship represented by the table shown below.

In the table above, a constant change of +1 in x corresponds to an increase in y by a constant factor of 4.

Therefore, the relationship given in the table above represents exponential growth, because each y-value is 4 times the value before it.

The points from this table lie on a smooth curve.

Linear vs Exponential Growth

Examples 1-2 : Tell whether each set of ordered pairs represents linear growth. Explain.

Example 1 :

{(2, 1), (5, 2), (8, 3), (11, 4)}

Solution :

Write the ordered pairs in a table and look for a pattern.

A constant change of +3 in x corresponds to a constant change of +1 in y.

Hence, the given set of ordered pairs represents linear growth.

Example 2 :

{(2, 1), (5, 2), (8, 3), (11, 4)}

Solution :

Write the ordered pairs in a table and look for a pattern.

A constant change of +5 in x corresponds to different changes in y.

The given set of ordered pairs does not represent linear growth.

Examples 3-4 : Tell whether each set of ordered pairs represents exponential growth. Explain.

Example 3 :

{(-1, 1.5), (0, 3), (1, 6), (2, 12)}

Solution :

Write the ordered pairs in a table and look for a pattern.

A constant change of +1 in x corresponds to an increase in y by a constant factor of 2.

Hence, the given set of ordered pairs represents exponential growth.

Example 4 :

{(-1, -9), (1, 9), (3, 27), (5, 45)}

Solution :

Write the ordered pairs in a table and look for a pattern.

A constant change of +2 in x corresponds to an increase in y, but NOT by a constant factor.

The given set of ordered pairs does not represent exponential growth.

Example 5 :

Check whether the following equation represents a linear growth.

y = 2x + 3

Solution :

Substitute values for x with constant difference, say

x = 1, 2, 3, 4

Substitute those values of x in the given equation and evaluate the values of y.

When x = 1,

y = 2(1) + 3

= 2 + 3

= 5

When x = 2,

y = 2(2) + 3

= 4 + 3

= 7

When x = 3,

y = 2(3) + 3

= 6 + 3

= 9

When x = 4,

y = 2(4) + 3

= 8 + 3

= 11

When x = 1, 2, 3, 4,

y = 5, 7, 9, 11

A constant change of +1 in x corresponds to a constant change of +2 in y.

Hence, the given equation represents a linear growth.

Example 6 :

Check whether the following equation represents a linear growth.

y = 2(3)^x

Solution :

Substitute values for x with constant difference, say

x = 0, 1, 2, 3

Substitute those values of x in the given equation and evaluate the values of y.

When x = 0,

y = 2(3)⁰

= 2(1)

= 2

When x = 1,

y = 2(3)¹

= 2(3)

= 6

When x = 2,

y = 2(3)²

= 2(9)

= 18

When x = 3,

y = 2(3)³

= 2(27)

= 54

When x = 0, 1, 2, 3,

y = 2, 6, 18, 54

A constant change of +1 in x corresponds to an increase in y by a constant factor of 3.

Hence, the given equation represents an exponential growth.

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