1

2 Any function of the form y = f (x) = ax 2 + bx + c where a  0 is called a Quadratic Function

3 Example: f(x) = 3x 2 – 2x – 5 a = 3, b = -2, c = -5 Note that if a = 0 we simply have the linear function f(x) = bx + c f(x) = bx + c Consider the quadratic function in general form:

4 Consider the simplest quadratic equation y = x 2 y = x 2 Here a = 1, b = 0, c = 0 Plotting some ordered pairs (x, y) we have: y = f (x ) = x 2 y = f (x ) = x 2 x f (x ) (x, y ) -3 9 (-3, 9) -2 4 (-2, 4) -1 1 (-1, 1) 0 0 (0, 0) 1 1 (1, 1) 1 1 (1, 1) 2 4 (2, 4) 2 4 (2, 4) 3 9 (3, 9) 3 9 (3, 9)

5 (x, y) (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9) x y (-2, 4) Vertex (0, 0) (2, 4) y = x 2 A parabola with the y-axis as the axis of symmetry. (1, 1) (-1, 1)

6 Graphs of y = ax 2 will have similar form and the value of the coefficient ‘a ’ determines the graph’s shape. x y y = x 2 y = 2x 2 y = 1 / 2 x 2 a > 0 opening up

7 x y y = -2x 2 a < 0 opening down In general the quadratic term ax 2in the quadratic function f (x (x ) = ax 2 +bx +bx + c determines the way the graph opens.

8 It turns out that the details of a quadratic function can be found by considering its coefficients a, b and c as follows: (1) Opening up (a > 0), down (a 0), down (a < 0) (2) y –intercept: c (3) x -intercepts from solution of y = ax 2 + bx + c = 0 y = ax 2 + bx + c = 0 (4) vertex = You solve by factoring or the quadratic formula

9 Example: y = f (x ) = x 2 - x - 2 here a = 1, b = -1 and c = -2 (1) opens upwards since a > 0 (2) y –intercept: -2 (3) x -intercepts from x 2 - x - 2 = 0 or (x -2)(x +1) = 0 or (x -2)(x +1) = 0 x = 2 or x = -1 x = 2 or x = -1 (4) vertex: Plug in x = to find the y-value

10 x y (-1, 0) (2, 0) y = x 2 - x - 2

11 Example: y = j (x ) = x here a = 1, b = 0 and c = -9 (1) opens upwards since a > 0 (2) y –intercept: -9 (3) x -intercepts from x = 0 or x 2 = 9  x =  3 or x 2 = 9  x =  3 (4) vertex at (0, -9)

12 x y (-3, 0) (3, 0) y = x (0, -9)

13 Example: y = g (x ) = x 2 - 6x + 9 here a = 1, b = -6 and c = 9 (1) opens upwards since a > 0 (2) y –intercept: 9 (3) x -intercepts from x 2 - 6x + 9 = 0 or (x - 3)(x - 3) = 0  x = 3 only or (x - 3)(x - 3) = 0  x = 3 only (4) vertex:

14 x y 3 9 (3, 0) (0, 9) y = x 2 - 6x + 9

15 Example: y = f (x ) = -3x 2 + 6x - 4 here a = -3, b = 6 and c = -4 (1) opens downwards since a < 0 (2) y –intercept: -4 (3) x -intercepts from -3x 2 + 6x - 4 = 0 (there are no x -intercepts here) (there are no x -intercepts here) (4) vertex at (1, -1) Vertex is below x-axis, and parabola opens down!

16 x y (1, -1) (0, -4) y = -3x 2 + 6x - 4

17 The Quadratic Formula It is not always easy to find x -intercepts by factoring ax 2 + bx + c when solving ax 2 + bx + c = 0 Quadratic equations of this form can be solved for x using the formula:

18 Example: Solve x 2 − 6x + 9 = 0 here a = 1, b = -6 and c = 9 as found previously Note: the expression inside the radical is called the “discriminant” Note: discriminant = 0 one solution

19 Example: Solve x 2 - x - 2 = 0 here a = 1, b = -1 and c = -2 Note: discriminant > 0 two solutions

20 Example: Find x -intercepts of y = x Solve x = 0 a = 1, b = 0, c = -9 x = 3 or x = -3 Note: discriminant > 0 two solutions

21 Example: Find the x -intercepts of y = f (x) = -3x 2 + 6x - 4 y = f (x) = -3x 2 + 6x - 4 a = -3, b = 6 and c = -4 Solve -3x 2 + 6x - 4 = 0  there are no x -intercepts as we discovered in an earlier plot of y = -3x 2 + 6x - 4 Note: discriminant < 0 no Real solutions

22 The end.