Hey there! Today we are going to explore how some things in our world grow incredibly fast using exponential functions.
Imagine you have a single magical cell that splits into two cells every second. Instead of adding a constant amount, we are multiplying by 2 over and over again!
This rapid doubling is called exponential growth. In just 4 steps, our single cell multiplies: 1 becomes 2, then 4, then 8, and finally 16!
When we graph this relationship, it starts out very flat and then suddenly curves almost straight up. The variable is in the exponent, like y = 2^x.
✏️ Worked example
If a colony of bacteria starts with 5 cells and doubles every hour, how many bacteria will there be after 3 hours?
Identify the starting amount (initial value), which is 5.
Identify the growth factor. Since it doubles, the growth factor is 2.
Set up the exponential equation: y = 5 * (2^x), where x is the number of hours.
Substitute x = 3 into the equation: y = 5 * (2³).
Calculate 2 cubed: 2 * 2 * 2 = 8.
Multiply by the starting amount: 5 * 8 = 40 bacteria.
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