# How do you write an exponential function whose graph passes through (0,0.2) and (4, 51.2)?

Dec 26, 2016

$f \left(x\right) = 0.2 \cdot {4}^{x}$

#### Explanation:

Notice that:

$\frac{51.2}{0.2} = 256 = {4}^{4}$

So we can write:

$f \left(x\right) = 0.2 \cdot {4}^{x}$

Then:

$f \left(0\right) = 0.2 \cdot {4}^{0} = 0.2$

$f \left(4\right) = 0.2 \cdot {4}^{4} = 51.2$

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Note

This is not the only solution.

Instead of $4$ we could use any of the other three $4$th roots of $256$, namely:

$- 4$, $4 i$ or $- 4 i$

So other solutions are:

$f \left(x\right) = 0.2 \cdot {\left(- 4\right)}^{x}$

$f \left(x\right) = 0.2 \cdot {\left(4 i\right)}^{x}$

$f \left(x\right) = 0.2 \cdot {\left(- 4 i\right)}^{x}$

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General case

Suppose we want an exponential function that passes through points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$

A general form can be written:

$f \left(x\right) = a \cdot {b}^{x}$

Then:

$f \left({x}_{1}\right) = a \cdot {b}^{{x}_{1}} = {y}_{1}$

$f \left({x}_{2}\right) = a \cdot {b}^{{x}_{2}} = {y}_{2}$

So:

${b}^{{x}_{2} - {x}_{1}} = \frac{a \cdot {b}^{{x}_{2}}}{a \cdot {b}^{{x}_{1}}} = {y}_{2} / {y}_{1}$

One solution is:

$b = {\left({y}_{2} / {y}_{1}\right)}^{\frac{1}{{x}_{2} - {x}_{1}}}$

There are other solutions formed by multiplying this by some $\left({x}_{2} - {x}_{1}\right)$th root of $1$ - that is some number of the form:

$\cos \left(\frac{2 k \pi}{{x}_{2} - {x}_{1}}\right) + i \sin \left(\frac{2 k \pi}{{x}_{2} - {x}_{1}}\right)$

where $k$ is any integer.

If ${x}_{2} - {x}_{1}$ is rational then this results in a finite number of other solutions.

If ${x}_{2} - {x}_{1}$ is irrational then this results in an infinite number of solutions.

Once we have a value for $b$ then there is a corresponding value for $a$ given by:

$a = {y}_{1} / {b}^{{x}_{1}}$