An exponential function passes through the points $(- 3, 8)$ $(-3, 8)$ and $(0, 1)$ $(0, 1)$ . What is the equation of the function?

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Answer 1 · 2017-03-02T00:22:53

The function has equation $y = {(\frac{1}{2})}^{x}$ $y = (1/2)^x$ .

It may look like you only have $1$ $1$ point, but in fact you have $2$ $2$ .

Every exponential function has its y-intercept at $(0, 1)$ $(0, 1)$ , because $n^{0} = 1$ $n^0 = 1$ for every real value of $n$ $n$ .

We conclude the equation can be written in the form

$y = a b^{x}$ $y = ab^x$

We can write a system of equations in two variables to solve for parameters $a$ $a$ and $b$ $b$ .

${(8 = ab^-3), (1 = ab^0):}$

We can see by inspection, using the property $n^0 = 1$ , that $a = 1$ .

We can now readily solve for $b$ .

$8 = 1(b^-3)$

$8 = b^-3$

$2^3 = b^-3$

$2^3 = 1/b^3$

$b^3 = 1/2^3$

$(b^(3))^(1/3) = (1/2^3)^(1/3)$

$b = 1/2$

The function therefore has equation $y = (1/2)^x$

Practice Exercises

a) It passes through the points $(2, 12)$ and $(-1, 3/2)$ .

b) The following graph:
enter image source here

Solutions
1a) $y = 3(2)^x$
1b) $y = 9(1/3)^x$

Hopefully this helps!

An exponential function passes through the points (-3, 8)(−3,8)(-3, 8) and (0, 1)(0,1)(0, 1). What is the equation of the function?