Question

Draw the exponential which passes through the points (−1, 8) and (3, 3) .
From your graph, estimate the halving time from your graph. Find an explicit formula, of the form f(x) = Aa^x
for the exponential and calculate the halving time exactly from this formula.

Answer 1

\[f(x)= A \alpha ^{x}\]

Answer 2

all that is needed are the values for A and \(\large \alpha \), so find them:

using (3, 3): \(\large 3=A \cdot \alpha ^{3} \)
using (-1, 8): \(\large 8=A \cdot \alpha^{-1} \)

solve this system of equations... can you do that?

Answer 3

I am not sure how to proceed.

Answer 4

solve for A in the first equation: \(\large 3=A \cdot \alpha^{3} \rightarrow A=\frac{3}{\alpha^3} \)

plug this expression into the second equation... can you do the substitution from here?

Answer 5

Still confused by the exponent being divided into alpha^3, what happens here?

Answer 6

substitute this into the second equation:
\(\large 8 = (\frac{3}{\alpha^3})\cdot \alpha^{-1} \)
\(\large 8 = (\frac{3}{\alpha^3})\cdot \frac{1}{\alpha} \)
\(\large 8 = \frac{3}{\alpha^4} \)
\(\large \alpha^4 = \frac{3}{8}\rightarrow \alpha=\sqrt[4]{\frac{3}{8}} \)

now that you've found alpha plug this in to the expression to find A.