1. For a quadratic function, how to find the coordinates of the vertex of its graph? 2. Functions and Graphs (a) For y = a(x - h) 2 +k, the vertex of the graph is (h, k). (b) For y = ax 2 +bx+c, the vertex of the graph is (, ). = a(x 2 + ) +c = a[x 2 + + - ] +c = a(x+ ) 2 - +c = a[x - ] 2 + k = ∴ h =, y-coordinate of Vertex Easy Memory Tips: Since ∆=b 2 - 4ac, we can memorize as. y = ax 2 +bx+c Take out the coefficient of x 2 in the first two terms Add up the square of half the coefficient of x Move outside the brackets By completing square, we can rewrite y = ax 2 +bx+c in the form of y = a(x - h) 2 +k, so as to find h and k. k h -

2. For y = a(x - h) 2 +k, the vertex of the graph is (h, k), whereE.g. y = x 2 - 12x+ ∴ The coordinates of the vertex of its graph are (6, 4). Completing Square = x 2 - 2(6)x+6 2 = (x - 6) 2 +4+4 k 36 +440 = x 2 - 12x+36+4 +4 h k = h =,. 2. Functions and Graphs For a quadratic function, how to find the coordinates of the vertex of its graph? For the function y = x 2 - 12x+40, find the coordinates of the vertex of its graph.

3. E.g. x-coordinate = = = y-coordinate = == ∴ The coordinates of the vertex of its graph are (6, 4). Algebraic Method The answer is the same as using completing square, but using the formula is more efficient. 2. Functions and Graphs For y = a(x - h) 2 +k, the vertex of the graph is (h, k), where k = h =,. For a quadratic function, how to find the coordinates of the vertex of its graph?

4. E.g. For y = 2x 2 +2x - 1, find the coordinates of the vertex of its graph. y = 2x 2 +2x - 1 = 2(x 2 +x) - 1 = 2[x 2 +x+ = 2(x+ ) 2 - ∴ The coordinates of the vertex of its graph Take out the coefficient of x 2 in the first two terms Add up the square of half the coefficient of x Take out from the bracket 2 ] = 2[x - ] 2 - k h are (, ). 2 + [ ] -1 2. Functions and Graphs Completing Square For y = a(x - h) 2 +k, the vertex of the graph is (h, k), where k = h =,. For a quadratic function, how to find the coordinates of the vertex of its graph?

5. E.g. For y = 2x 2 +2x - 1, find the coordinates of the vertex of its graph. x-coordinate = == y-coordinate === ∴ The coordinates of the vertex of the graph are (, ). 2. Functions and Graphs Algebraic Method The answer is the same as using completing square, but using the formula is more efficient. For y = a(x - h) 2 +k, the vertex of the graph is (h, k), where k = h =,. For a quadratic function, how to find the coordinates of the vertex of its graph?

6. E.g. x-coordinate = == ∴ The coordinates of the vertex of the graph are (, ). It is hard to memorize the formula for the y-coordinate. We can find the x-coordinate first and then substitute it into the equation to find the y-coordinate. - 1 - 1 2x 2 - 2x+1 Substitute x= into the equation, we have 2( ) 2 +2( ) - 1 y = 2. Functions and Graphs Algebraic Method For y = 2x 2 +2x - 1, find the coordinates of the vertex of its graph. For y = a(x - h) 2 +k, the vertex of the graph is (h, k), where k = h =,. For a quadratic function, how to find the coordinates of the vertex of its graph?