Quadratic Functions in the Form y = a(x – h)2 + k

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Presentation transcript:

Quadratic Functions in the Form y = a(x – h)2 + k

Can we obtain the features of the graph of a quadratic function without plotting its graph? Yes. For the function y = a(x – h)2 + k, we can obtain the axis of symmetry and the coordinates of the vertex directly.

Let us study quadratic functions in the form y = a(x – h)2 + k. Can we obtain the features of the graph of a quadratic function without plotting its graph? Let us study quadratic functions in the form y = a(x – h)2 + k.

y = a(x – h)2 + k Case 1 : a > 0 a > 0 and (x – h)2  0 O y x a > 0 and (x – h)2  0 a(x – h)2  0 a(x – h)2 + k  k

The minimum value of y is k when x = h. y = a(x – h)2 + k axis of symmetry: x = h Case 1 : a > 0 O y x a > 0 and (x – h)2  0 a(x – h)2  0 y  k minimum value of y The minimum value of y is k when x = h. k vertex: (h, k)

y = a(x – h)2 + k Case 2 : a < 0 a < 0 and (x – h)2  0 O y x a < 0 and (x – h)2  0 a(x – h)2  0 a(x – h)2 + k  k

The maximum value of y is k when x = h. y = a(x – h)2 + k vertex: (h, k) Case 2 : a < 0 O y x a < 0 and (x – h)2  0 maximum value of y k a(x – h)2  0 y  k The maximum value of y is k when x = h. axis of symmetry: x = h

Maximum or minimum value Features of the function y = a(x – h)2 + k and its graph a > 0 a < 0 Direction of opening Axis of symmetry Vertex Maximum or minimum value upwards downwards x = h (h, k) (min. point) (max. point) min. value = k max. value = k

∵ Coefficient of x2 = 1 > 0 Can you find the minimum value of the function y = (x + 5)2 + 7 and the axis of symmetry of its graph? y = a(x – h)2 + k y = (x + 5)2 + 7 = [x – (–5)]2 + 7 ◄ a = 1, h = –5, k = 7 ∵ Coefficient of x2 = 1 > 0 min. value = k ∴ The minimum value of y is 7. ∴ The axis of symmetry is x = –5. x = h

Follow-up question For the function y = 9 – 2(x + 1)2, find (a) its maximum or minimum value, (b) the coordinates of the vertex. (a) y = 9 – 2(x + 1)2 = –2(x + 1)2 + 9 For y = a(x – h)2 + k, coordinates of the vertex: (h, k) = –2[x – (–1)]2 + 9 ∵ Coefficient of x2 = –2 < 0 ∴ The maximum value of y is 9. (b) Coordinates of the vertex = (–1, 9)