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Write an exponential function in the form y, equals, a, b, to the power xy=ab x that goes through the points left bracket, 0, comma, 20, right bracket(0,20) and left bracket, 3, comma, 2500, right bracket(3,2500).

Question

Write an exponential function in the form y, equals, a, b, to the power xy=ab x that goes through the points left bracket, 0, comma, 20, right bracket(0,20) and left bracket, 3, comma, 2500, right bracket(3,2500).

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Solution

The general form of an exponential function is y = ab^x.

We know that the function passes through the points (0,20) and (3,2500).

We can use these points to find the values of a and b.

  1. Substitute the coordinates of the first point into the equation:

    20 = ab^0

    Since any number (except 0) to the power of 0 is 1, we get a = 20.

  2. Substitute a = 20 and the coordinates of the second point into the equation:

    2500 = 20b^3

    Divide both sides by 20 to solve for b:

    b^3 = 2500 / 20 = 125

    Take the cube root of both sides to solve for b:

    b = ∛125 = 5

So, the exponential function that goes through the points (0,20) and (3,2500) is y = 20 * 5^x.

This problem has been solved

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