f(x)=−3x 2 +ax+bFor the quadratic function given above, a and b are constants such the x-coordinate of the vertex is 2.5. If the graph of the function given above is tangential to the line y=10, then find the value of f(3).
Question
f(x)=−3x 2 +ax+bFor the quadratic function given above, a and b are constants such the x-coordinate of the vertex is 2.5. If the graph of the function given above is tangential to the line y=10, then find the value of f(3).
Solution
The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Given that the x-coordinate of the vertex is 2.5, we can rewrite the given function as:
f(x) = -3(x - 2.5)^2 + k
The graph of the function is tangential to the line y = 10, which means the vertex of the parabola is on the line y = 10. Therefore, we can set k = 10:
f(x) = -3(x - 2.5)^2 + 10
Now we can find the value of f(3) by substituting x = 3 into the equation:
f(3) = -3(3 - 2.5)^2 + 10 = -3(0.5)^2 + 10 = -3(0.25) + 10 = -0.75 + 10 = 9.25
So, the value of f(3) is 9.25.
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