exponential growth
big idea- growth at a constant percentage rate can be described by an expression of the form bg^x , where g > 1 and the variable is in the exponent
What is exponential growth?
In exponentaial growth, the original amount is repeatedly mulitipled by a postive number called the growth factor.
growth model for powering
If a quanity is multipled by a postive number g ( the growth factor) in each of x time periods then, after the x periods, the quanity will be multiplyed by g^x.
what happens if the exponent is zero
In the exponential growth equation y = b x g^x, x can be any real number. Consider the situation when x = 0. In the 0 time periods no time has elapsed . The starting amoun b has not grown at all and so it remains the same. It can remain the sameonly if it is multiplied by 1. This means hat g^0 = 1 , regardless of the value of the growth factor g. This property applies also when g is a negatve number.
zero exponent property
If x is any nonzero real number, then x^0 = 1
what does a graph of exponential growth look like
An equation of the form y = b x g ^x , where g is a number greater than 1, can describe expoential growth. Graphs of such equations are not lines. They are exponential growth curves.
example 1
Solution- Substitute x = 0, 1, 2, 3, and 4 into the formula y = 1.5 x 2^x . Below we show the computation and the results listed as (x,y) pairs.
Comptation (x,y)
1.5 x 2^0 = 1.5 x 1 = 1.5 (0, 1.5)
1.5 x 2^1 = 1.5 x 2 = 3 (1, 3)
1.5 x 2^2 = 1.5 x 4 = 6 (2, 6)
1.5 x 2^3 = 1.5 x 8 = 12 (3, 12)
1.5 x 2^4 = 1.5 x 16 = 24 (4, 24)
Comptation (x,y)
1.5 x 2^0 = 1.5 x 1 = 1.5 (0, 1.5)
1.5 x 2^1 = 1.5 x 2 = 3 (1, 3)
1.5 x 2^2 = 1.5 x 4 = 6 (2, 6)
1.5 x 2^3 = 1.5 x 8 = 12 (3, 12)
1.5 x 2^4 = 1.5 x 16 = 24 (4, 24)
exponential population grwoth
Other than money calculated using comppound interest, few things in the real world grow exactly exponetially. However, exponential groth curves can be used to approximate changes in population. .
Vocabulary
Exponential Growth- A situation in which y = bg^x and g > 1
Growth Factor- In exponentila growth or decay, the positive number which is repeatedly multipied by the original amount.
Exponential Growth Equation-If the amount at the beginning of the growth period is b, the growth factor is g, and y is the amount after x time periods, then y = b x g^x
Growth Factor- In exponentila growth or decay, the positive number which is repeatedly multipied by the original amount.
Exponential Growth Equation-If the amount at the beginning of the growth period is b, the growth factor is g, and y is the amount after x time periods, then y = b x g^x