Given the exponential function:
[tex]f(x)=a\cdot b^x[/tex]The function passes through the points:
(0, 4) and (3, 500)
Let's find the values of a and b.
Let's use the given points to find a and b.
(0, 4):
Substitute 0 for x, and 4 for f(x), to find the value of a:
[tex]\begin{gathered} f(x)=a\cdot b^x \\ \\ 4=a\cdot b^0 \\ \\ 4=a\cdot1 \\ \\ 4=a \\ \\ a=4 \end{gathered}[/tex]The value of a is 4
(3, 500):
Substitute 3 for x, 500 for f(x) and 4 for a to solve for b:
[tex]\begin{gathered} f(x)=a\cdot b^x \\ \\ 500=4\cdot b^3 \\ \\ \text{Divide both sides by 4:} \\ \frac{500}{4}=\frac{4\cdot b^3}{4} \\ \\ 125=b^3 \\ \\ \text{Take the cube root of both sides:} \\ \sqrt[3]{125}=\sqrt[3]{b^3} \\ \\ 5=b \\ \\ b=5 \end{gathered}[/tex]The value of b is 5
ANSWER:
a = 4
b = 5
