1. Quadratic functions
The general form of a quadratic is: y = ax2 + bx + c
A more basic form of this equation is:
y = x2 or y = ax2
If a > 0 (or positive) the parabola opens up, or is happy.
y = x2
y = 2x2
y = x2
y = ½x2 x -3 -2 -1 1 2 3 y 9 4 1 1 4 9
y = 2x2 x -3 -2 -1 1 2 3 y 18 8 2 2 8 18
y = ½x2 x -4 -2 -1 1 2 4 y 8 2 2 8

2. Quadratic functions y = ax2
If a < 0 (or negative) the parabola opens down, or is sad.
y = -x2
y = -½x2 x -3 -2 -1 1 2 3 y -9 -4 -1 -1 -4 -9
y = -2x2 x -3 -2 -1 1 2 3 y
y = -x2
-18 -8 -2 -2 -8
-18
y = -½x2 x -4 -2 -1 1 2 4 y
y = -2x2 -8 -2 -½ -½ -2 -8

3. Quadratic functions y = ax2 ± c
y = 4  x2 x -3 -2 -1 1 2 3 y
y = 2x2  8 -5 3 4 3 -5
As a < 0 (or negative) the parabola opens down, or is sad.
The constant term, +4, is how far the curve has been moved up.
y = 2x2  8 x -3 -2 -1 1 2 3 y 10 -6 -8 -6 10
As a > 0 (or positive) the parabola opens up, or is happy.
The constant term, 8, is how far the curve has been moved down.
y = 4  x2

4. Quadratic functions y = ax2 + bx + c
y = x2 + 2x  3
(1,9)
y = x2 +2x3 x -4 -3 -2 -1 1 2 y 5 -3 -4 -3 5
x = 1
As a > 0 the parabola opens up.
The constant term is 3 and this is the y-intercept.
y = 8 + 2x  x2
x = -1 x -3 -2 -1 1 2 3 4 5 y -7 5 8 9 8 5 -7
As a < 0 the parabola opens down.
The constant term is +8 and this is the y-intercept.
(-1,-4)
y = 8+2x x2

5. Features of a quadratic function
Parabolas start and finish in the same direction.
Parabolas are symmetrical.
The turning point of a parabola is called a vertex.
If a > 0, the parabola is concave up or happy.
If a < 0, the parabola is concave down or sad.
On the previous slides we saw that if:
the larger a was the steeper the curve,
the smaller a was the flatter the curve.
The graph of y = ax2 + c is the parabola y = ax2 moved up c units.
The graph of y = ax2  c is the parabola y = ax2 moved down c units.
The graph of y = ax2 + bx + c is the parabola cuts the y-axis at c.
The axis of symmetry is halfway between the two x-intercepts.
The axis of symmetry runs through the vertex.
vertex

6. Today’s work
Exercise 12C page 366
#1, 6, 8ab, 9, 10, 11