Question 200454
Exponential functions are in the form: {{{y = k*a^x}}} where k and a are constants. So our task is to find the values of k and a.<br>
We can do this by using the points we know will fit the equation: (1, 3) and (3, 4.32). Let's start with the first point. Substituting the x and y coordinates from (1, 3) for the x and y in {{{y = k*a^x}}} we get:
{{{(3) = k*a^(1)}}}
Simplifying we get:
{{{3 = k*a}}}
Dividing both sides by a we get:
{{{3/a = k}}}<br>
Now let's do it with the second point. We will use the x and y from (3, 4.32) <b>and</b> the "value" we just got for k: 3/a. Substituting these 3 values into {{{y = k*a^x}}} we get:
{{{(4.32) = (3/a)*a^3}}}
Simplifying we get:
{{{4.32 = 3a^2}}}
Dividing both sides by 3 we get:
{{{1.44 = a^2}}}
Finding the square root of both sides we get:
{{{1.2 = a}}} or {{{-1.2 = a}}}
Since we usually want a > 0 then we <i>may</i> reject the a = -1.2.
As a fraction 1.2 is 6/5. If a = 6/5 then k = 3/(6/5) = 5/2. So our exponential equation is:
{{{y = (5/2)(6/5)^x}}} or ,as decimals, {{{y = 2.5(1.2)^x}}})