How do you find an exponential function given the points are (-1,8) and (1,2)?

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How do you find an exponential function given the points are (-1,8) and (1,2)?

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Answer 1 · 2016-01-09T00:37:31

$y = 4 {(\frac{1}{2})}^{x}$ $y=4(1/2)^x$

Explanation:

An exponential function is in the general form

$y = a {(b)}^{x}$ $y=a(b)^x$

We know the points $(- 1, 8)$ $(-1,8)$ and $(1, 2)$ $(1,2)$ , so the following are true:

$8 = a (b^{- 1}) = \frac{a}{b}$ $8=a(b^-1)=a/b$

$2 = a (b^{1}) = a b$ $2=a(b^1)=ab$

Multiply both sides of the first equation by $b$ $b$ to find that

$8 b = a$ $8b=a$

Plug this into the second equation and solve for $b$ $b$ :

$2 = (8 b) b$ $2=(8b)b$

$2 = 8 b^{2}$ $2=8b^2$

$b^{2} = \frac{1}{4}$ $b^2=1/4$

$b = \pm \frac{1}{2}$ $b=+-1/2$

Two equations seem to be possible here. Plug both values of $b$ $b$ into the either equation to find $a$ $a$ . I'll use the second equation for simpler algebra.

If $b = \frac{1}{2}$ $b=1/2$ :

$2 = a (\frac{1}{2})$ $2=a(1/2)$

$a = 4$ $a=4$

Giving us the equation: $y = 4 {(\frac{1}{2})}^{x}$ $color(green)(y=4(1/2)^x$

If $b = - \frac{1}{2}$ $b=-1/2$ :

$2 = a (- \frac{1}{2})$ $2=a(-1/2)$

$a = - 4$ $a=-4$

Giving us the equation: $y = - 4 {(- \frac{1}{2})}^{x}$ $y=-4(-1/2)^x$

However! In an exponential function, $b > 0$ $b>0$ , otherwise many issues arise when trying to graph the function.

The only valid function is

$y = 4 {(\frac{1}{2})}^{x}$ $color(green)(y=4(1/2)^x$

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How do you find an exponential function given the points are (-1,8) and (1,2)?

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