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

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

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Answer 1 · 2016-12-26T13:45:08

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

Explanation:

Notice that:

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

So we can write:

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

Then:

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

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

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Note

This is not the only solution.

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

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

So other solutions are:

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

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

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

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General case

Suppose we want an exponential function that passes through points $(x_{1}, y_{1})$ $(x_1, y_1)$ and $(x_{2}, y_{2})$ $(x_2, y_2)$

A general form can be written:

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

Then:

$f (x_{1}) = a \cdot b_{1}^{x_{1}} = y_{1}$ $f(x_1) = a*b^(x_1) = y_1$

$f (x_{2}) = a \cdot b_{2}^{x_{2}} = y_{2}$ $f(x_2) = a*b^(x_2) = y_2$

So:

$b_{2}^{x_{2} - x_{1}} = \frac{a \cdot b_{2}^{x_{2}}}{a \cdot b_{1}^{x_{1}}} = \frac{y_{2}}{y_{1}}$ $b^(x_2-x_1) = (a*b^(x_2))/(a*b^(x_1)) = y_2/y_1$

One solution is:

$b = {(\frac{y_{2}}{y_{1}})}^{\frac{1}{x_{2} - x_{1}}}$ $b = (y_2/y_1)^(1/(x_2-x_1))$

There are other solutions formed by multiplying this by some $(x_{2} - x_{1})$ $(x_2-x_1)$ th root of $1$ $1$ - that is some number of the form:

$cos (\frac{2 k π}{x_{2} - x_{1}}) + i sin (\frac{2 k π}{x_{2} - x_{1}})$ $cos((2kpi)/(x_2-x_1)) + i sin((2kpi)/(x_2-x_1))$

where $k$ $k$ is any integer.

If $x_{2} - x_{1}$ $x_2-x_1$ is rational then this results in a finite number of other solutions.

If $x_{2} - x_{1}$ $x_2-x_1$ is irrational then this results in an infinite number of solutions.

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

$a = \frac{y_{1}}{b_{1}^{x_{1}}}$ $a = y_1/b^(x_1)$

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

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Explanation:

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