Question

Find a formula for the exponential function of the form = ^ that passes through the points (−4, 19) and (7, 3).Round the value of the base of the exponential to 4 decimal places

Answer 1

We are given that the graph of the function passes through (-4,19), and (7,3), therefore:

$\begin{gathered} 19=ab^{-4}, \\ 3=ab^7. \end{gathered}$

Dividing the second equation by the first one, we get:

$\begin{gathered} \frac{3}{19}=\frac{ab^7}{ab^{-4}}=\frac{3}{19}=b^{7+4}, \\ b^{11}=\frac{3}{19}, \\ b=\sqrt[11]{\frac{3}{19}}. \end{gathered}$

Now, substituting the above result in the second equation, we get:

$\begin{gathered} a(\sqrt[11]{\frac{3}{19}})=3, \\ a=\frac{3}{(\sqrt[11]{\frac{3}{19}})^7}. \end{gathered}$

Rounding b to 4 decimal places, we get:

$\begin{gathered} \\ b\approx0.8455. \end{gathered}$

Answer:

$\begin{gathered} a=\frac{3}{(\sqrt[11]{\frac{3}{19}})^7}, \\ b=0.8455. \end{gathered}$