-e^[(-(x-a)^2)/b] b>0; local max at x=7 and points of inflection are x=5 and x=9. Find the formula...
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To find the formula for the given function, we can start by identifying its key characteristics - a local maximum at x = 7 and points of inflection at x = 5 and x = 9.
Let's break down the steps to find the formula:
Step 1: Start with the basic exponential function - e^x.
Step 2: Introduce the first transformation, which is a horizontal shift to the right by 'a' units. This can be achieved by replacing 'x' with '(x - a)' in the exponential function. We now have e^(x - a).
Step 3: Apply the second transformation, which is a reflection about the x-axis, resulting in a negative exponent. Replace 'x' in the equation with '(x - a)' and square it to achieve a parabolic shape. We then have e^(-(x - a)^2).
Step 4: Finally, apply the third transformation, which is a horizontal compression/horizontal stretching of the parabolic shape. The 'b' in the exponent controls the compression/stretching. We introduce 'b' in the exponent as well, resulting in e^(-(x - a)^2/b).
Combining all the steps, we have the formula: y = -e^(-(x - a)^2/b), where a is the horizontal shift, and b is the horizontal compression factor.
Therefore, the formula for the given function is:
y = -e^(-(x - 7)^2/b), where b > 0.