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.
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.
Remember that linear functions have constant differences. Exponential functions do not have constant differences, but they do have constant ratios.
In an exponential function, as the x-values increase by a constant amount, the y-values are multiplied a constant amount. This amount is the constant ratio and is the value of b in f(x) = ab^{x}.
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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