Explanation:
We can find the parameters of the exponential equation ...
y = a^(x-h) +k
1. Locate the horizontal asymptote. The y-value of that is the value of k.
2. Locate the point that is at y = k+1. The x-value of that is the value of h.
3. Locate the point that is at x = h+1. Subtract k from the y-value of that to find "a".
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You have identified the three points on any exponential curve y = a^(x -h) +k. They are ...
(-∞, k) for growth, or (∞, k) for decay
(h, k+1)
(h+1, k+a) . . . . . for decay, (h -1, k+1/a) might be easier to find
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<em>Comment on reflected curves</em>
If the curve extends below the horizontal asymptote, it has been reflected about the x-axis. Reflect it across the x-axis, perform the above steps, then multiply the right side of the equation by -1.
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<em>Additional note</em>
If the value of h is not an integer, it might be easier to express the curve in the form ...
y = a·b^x +k
In this case, you can find the value of b by finding the y-values associated with three consecutive values of x. Call them (x, y1), (x+1, y2), (x+2, y3). Then ...
b = (y3 -y2)/(y2 -y1)
a = (y1 -k)/b^x . . . . . where x is the x-value of the first point, (x, y1)
