Question

Describe a method for writing an exponential equation given a graph

Answer 1

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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Comment on reflected curves

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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Additional note

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)