What is the y = A b^x form of the exponential function thru the points (1, 5) and (6, 9)?  What is the y = A e^(kx) form of this function?

Recall that to fit three points to a quadratic we substituted the coordinates of those points into the form y = a t^2 + b t + c of a quadratic.

Now we have two points.  What form do we fit to these points?

The requested form is y = A b^x.

Plugging in the coordinates of our points we get

5 = A b^1

9 = A b^6.

We can solve these equations by dividing the second equation by the first, obtaining

9/5 = (A b^6) / (A b^1), which we simplify to get

1.8 = b^5.  Taking the 1/5 power of both sides we have

(b^5)^(1/5) = 1.8^(1/5)

b = 1.125.

Now substituting b = 1.25 into the first equation we get

5 = A * 1.125^1 so that

A = 5 / 1.125 = 4.

Our y = A b^x function is therefore

y = 4 * 1.125^x.

The y = A e^(kx) form is obtained by setting b^x = e^(kx)

b^x = e^(kx) so

1.125^x = e^(kx).  Since e^(kx) = (e^k)^x we have

1.125^x = (e^k)^x so

e^k = 1.125.  Taking natural log of both sides we get

ln(e^k) = ln(1.125) or

k = ln(1.125) = .1178.

So the y = A e^(kx) form of the function y = 4 * 1.125^x is

y = 4 * e^(.1178 x) .

If f = 0 then

• f + g = g (point on the graph of g(x) function)
• f * g = 0 (point on the x axis)
• f / g = 0 (point on the x axis)
• g / f undefined (if g not zero then asymptote; on each side look at signs of f and g to determine sign of g / f)

If f = 1 then

• f + g is 1 unit higher than g
• f * g = g (point on graph of g(x))
• g / f = g (point on graph of g(x))

If f = -1 then

• f + g is 1 unit lower than f
• f * g = -g (point on graph of -g(x), reflection thru x axis of graph of g(x))
• g / f = -g (point on graph of -g(x))

Between consecutive zeros and/or asymptotes (if functions continuous between zeros and asymptotes):

• neither function can change sign (if functions are continuous)
• if f and g of like signs f g > 0

Recall last class we obtained the equation

T = A e^(k t) + Troom , A not zero.

If we know that the temperature of a glass of water is 100 degrees when t = 0 and 60 degrees when t = 20 minutes, while Troom is 40 degrees, what are the values of A and k?