Question

an exponential function ( f(x) = a cdot b^x ) passes through the points ( (0, 7) ) and ( (3, 56) ). what are the values of ( a ) and ( b )?( a = square ) and ( b = square )question help: (\boldsymbol{\text{video}})

Explanation:

Step1: Find $a$ using $(0,7)$

Substitute $x=0$, $f(x)=7$ into $f(x)=a\cdot b^x$:
$7 = a\cdot b^0$
Since $b^0=1$, we get $a=7$.

Step2: Substitute $a$ and $(3,56)$

Substitute $a=7$, $x=3$, $f(x)=56$ into the function:
$56 = 7\cdot b^3$

Step3: Solve for $b$

First, divide both sides by 7:
$\frac{56}{7}=b^3$
$8 = b^3$
Take the cube root of both sides:
$b=\sqrt[3]{8}=2$

Answer:

$a = 7$ and $b = 2$