The curve of a quadratic function f(x)=2(x-h)^2+2k intersects the x – axis at points (1, 0) and (5, 0). The straight line y = – 8 touches the minimum point of the curve.
(i) Find the value of h and k.
(ii) Hence, sketch the graph of f (x) showing clearly the minimum point and the axis intercepts.
(iii) If the graph is reflected about the x – axis, write the equation of the curve.
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Assume the quadratic equation as px^2 + qx + r = y.
Line passes through (-1, 0): ==> p -q + r = 0 —-> (1)
y’(x) = 2px +q; y’(1) = 1 = 2p + q. —-> (2)
Line y = x touches curve at (1, 1) ==> p + q + r = 1 —-> (3).
Eqs (1) & (3) ==> q = 1/2. From (2), p = 1/4. r = 1 - p - q = 1 - 1/4 - 1/2 = 1/4.
The quadratic equation is (1/4)x^2 + (1/2)x + (1/4) = y. ==> x^2 + 2x + 1 = y + 3/4.
==> (x + 1)^2 = (y + 3/4). Focal length = 1/4.
Axis: x = -1. Directrix: y = -1. Focus: (-1, -1/2)
Step-by-step explanation:
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