We're asked to determine an exponential function that passes through the points negative one, five-fourths, and two eighty. We also want to find the exponential growth or decay rate and the function value when x equals three. If we have an exponential function in this form here, the base b is equal to one plus r, where r is equal to the exponential growth or decay rate, expressed as a decimal. A is the initial value or initial amount, which is the function value when x equals zero.
and x is the exponent. So going back up to our question, if the function contains the point negative one 5 4ths, this means that f of negative one must equal 5 4ths, and if the function contains the point 2 80, this means f of two is equal to 80. Notice how we're not given the point where x equals zero. So we don't know the initial value or the value of a or the value of b, the base. So all we'll have to do is set this up as a system of equations and then solve for a and b. So for f of negative one, we'll replace x with negative one and set the function value equal to five-fourths.
So we would have a times b to the power of negative one equals five-fourths. And then if f of two equals eighty, that means A times B to the second must equal 80. So now we'll solve this as a system of equations. So let's call this equation one and this equation two.
So using equation one, we know A times B to the negative one equals 5 fourths. So what we'll do is solve this equation for A and then perform substitution. into the second equation.
And since we have b to the negative one power here, we can multiply both sides of the equation by b to the first. Remember when multiplying and the bases are the same, we add our exponents. Negative one plus one is zero. B to the zero would equal one. So we have a equals, this would be just 5 4ths b.
So now we'll substitute this for a in equation two. So now equation two is going to be five-fourths b times b to the second must equal 80. So we would have five-fourths b to the third equals 80. And now we can solve for b to the third by multiplying both sides by four-fifths. So let's go ahead and put eighty over one.
Notice on the left side, the four is simplified to one, the five is simplified to one. So we have b to the third equals, on the right side, the five and the eighty simplify. There's one five and five and sixteen fives and eighty.
Sixteen times four is equal to sixty-four. So we have b to the third equals sixty-four. And now to solve for b, we'll take the cube root of both sides.
Or if we want we could raise both sides to the 1 3rd power. Let's go ahead and take the cube root of both sides. So the cube root of b to the 3rd is b.
The cube root of 64 is equal to 4, since 4 to the 3rd does equal 64. So now we know b is equal to 4, we now have to find the value of a. So we know that f of x, must equal a times four to the power of x. Now we still have to find the value of a, or the initial value, and we can do this using the fact that f of negative one equals 5 fourths, or f of two equals 80. So let's go ahead and determine the value of a on the next slide.
So again, if we know that f of two equals 80, we can substitute two for x, and then set the function value equal to 80 to solve for a. So for f of two, we would have a times four to the second, which must equal 80. Well, four squared is 16, so we have 16a equals 80. Divide both sides by 16, we know that a must equal five. So the exponential function that passes through the two given points would be f of x equals a, which is five, times our base, which is four raised to the power of x. To check this, let's go ahead and graph this function to make sure it passes through these two points.
So here's our function graphed in blue. Notice how we have exponential growth and it does pass through the two given points. So our function is correct, but we're still asked to determine the exponential growth rate and to determine the function value when x equals three.
To determine the exponential growth rate, we need to remember that our base b is equal to one plus r, where r is the growth rate expressed as a decimal. So for our function, Since our base is equal to four, we would have four equals one plus r. Subtracting one on both sides, we have r equals three.
Well three as a percentage is three hundred percent, so the exponential growth rate is three hundred percent. Which means, each time x increases by one, the function value increases three hundred percent. And then for the last question, to determine the function value when x equals three, we need to find f of three, which is equal to five, times four raised to the third power. Well, four to the third is equal to 64, so we have five times 64, and five times 64 is equal to 320. So now we have our exponential function, the exponential growth rate, and the function value when x equals three. I hope you found this helpful.