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

2. 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.

3. 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.

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

5. 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)

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

7. 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

8. 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

9. ∵ 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

10. 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)