1. Math
426
FUNCTIONS
QUADRATIC

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

3. a b c y = 3x 2 - 2x + 1 = 3, = -2, = 1 Example:
Note that if a = 0 we simply have the linear function
y = bx + c

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

5. y = x2 (x, y) (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9)
Vertex (0, 0)
A parabola with the y-axis as the axis of symmetry.

6. Graphs of y = ax 2 will have similar form and the value of the coefficient ‘a ’ determines the graph’s shape. y
y = 2x 2 4 3 2 1
y = x 2
y = 1/2 x 2
a > 0
opening up x

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

8. Consider f (x ) = ax 2 +bx + c
In a general sense the linear term bx acts to shift the plot of f (x ) from side to side and the constant term c (=cx 0) acts to shift the plot up or down. y
a > 0
x-intercept c
Notice that c is the y -intercept where x = 0 and f (0) = c
a < 0 x c
y-intercept
Note also that the x -intercepts (if they exist) are obtained by solving: y = ax 2 +bx + c = 0

9. (1) Opening up (a > 0), down (a < 0)
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)
(2) y –intercept: c
(3) x -intercepts from solution of
y = ax 2 + bx + c = 0
You solve by factoring or the quadratic formula
(4) vertex =

10. 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
x = 2 or x = -1
(4) vertex:

11. y
(-1, 0)
(2, 0) x
y = x 2 - x - 2 -1 -2 -3

12. Example: y = j (x ) = x 2 - 9 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
(4) vertex at (0, -9)

13. y
(-3, 0)
(3, 0) x
y = x 2 - 9 -9
(0, -9)

14. 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
(4) vertex:

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

16. 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)
(4) vertex at (1, -1)
Vertex is below x-axis, and parabola opens down!

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

18. ax 2 + bx + c = 0 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:

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

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

21. Example: Find x -intercepts of y = x 2 - 9
Solve x = 0
a = 1, b = 0, c = -9
Note: discriminant > two solutions
x = 3 or x = -3

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

23. FUNCTIONS
QUADRATIC
The end.