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 ½ ½ 2 8

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

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

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

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

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