Welcome to an Introduction
to Exponential Equations in Two Variables.
An exponential equation is an equation
in the form y equals a times b raised to the power of x.
Notice how the variable x
is in a position of the exponent.
A is called the initial or starting value,
which is the y value when x is zero,
which also means zero comma a is the vertical intercept.
B is the base, which is also called
the growth factor or decay factor,
where if the base b is greater than one,
the graph is increasing
and we have exponential growth.
Here we see the graph
of y equals two raised to the power of x,
where a is equal to one and b is equal to two.
Since two is greater than one,
we have exponential growth.
Notice how the graph goes uphill from left to right.
If the base b is between zero and one,
the graph is decreasing and we have exponential decay.
Here we see the graph
of y equals 1/2 raised to the power of x,
which means a is equal to one and b is equal to 1/2.
Because 1/2 is between zero and one,
this is exponential decay.
Notice how the graph goes downhill from left to right,
or the graph is decreasing.
And if r is the percent growth
or percent decay as a decimal,
the equation can be written in one of the following forms.
If we have exponential growth,
the base b is equal to one plus r,
where r is the percent growth as a decimal.
And if we have exponential decay,
the base b is equal to one minus r,
where r is the percent of decay as a decimal.
Let's look at some examples.
We're asked to complete the table.
In these first two rows, we are given the equation
and asked to determine the initial value,
whether we have growth or decay,
and then determine the growth or decay rate as a percentage.
So the first equation is y equals 12
times 1.2 raised to the power of x.
Notice a is equal to 12
and therefore the initial value is 12.
Notice how the base b is 1.2, which is greater than one,
which means we have exponential growth.
And now to find the growth rate as a percentage,
remember when we have exponential growth,
the base b is equal to one plus r,
where r is the growth rate as a decimal.
If we solve this equation for r,
we have r equals b minus one,
where in our case the base b is 1.2.
So to find r, since we know r is equal to b minus 1
and the base b is 1.2, we have r equals 1.2 minus one,
giving us r equals 0.2.
But we're asked to express this as a percentage.
0.2 or 2/10 is equal to 20%.
So we have a 20% growth rate for this first equation.
For the next equation we have y equals five
times 0.85 raised to the power of x.
Notice a is equal to five.
Five is the initial value.
This time the base b is 0.85,
which is between zero and one,
and therefore we have exponential decay.
Because we have exponential decay,
the base is equal to one minus r.
If solve this equation for r,
we have r equals one minus b.
Because we have exponential decay,
in order to find the decay rate as a percentage,
we use the equation r equals one minus b,
where here the base b is 0.85.
So r is equal to one minus 0.85,
and therefore r is equal to 0.15,
which as a percentage is 15%.
Here we have a 15% decay rate.
So again, it's important to remember
when we have exponential growth, r is equal to b minus one.
If we have exponential decay, r is equal to one minus b.
In these last two rows, we are given the initial value,
whether we have growth or decay,
and then given the growth or decay rate,
and we're asked to find the equation.
So for this third row we have exponential growth,
so we'll be using these formulas here.
So we have y equals a
times the quantity one plus r raised to the power of x,
where a is the initial value.
So we know a is equal to 9.2.
We already know we have exponential growth
and the growth rate is 6.2%,
which we need to convert to a decimal.
So 6.2% is equal to 0.062.
So now we substitute a and r into our equation,
which gives us y equals 9.2, the initial value,
times the quantity one plus 0.062 raised to the power of x,
which gives us y equals 9.2
times 1.062 raised to the power of x for our equation.
And now for the last row, we have an initial value of 81.
So a is equal to 81.
We have exponential decay,
which means our equation is going to be in this form here,
where we have y equals a times the quantity one minus r
raised to the power of x.
And we're told the decay rate is 12.5%.
So r is equal to 12.5%,
which we have to convert to a decimal.
12.5% is equal to 0.125.
So the equation is y equals a, which is 81,
times the quantity one minus r,
giving us one minus 0.125 raised to the power of x.
Simplifying, we have y equals 81
times 0.875 raised to the power of x.
I hope you found this helpful.