Quadratic Equations A term like x2 is called a square in algebra because it is the area of a square with side x The adjective quadratic comes from the Latin word quadratum for square. So a quadratic equation is one in which the highest index number of a term with x in is x2 Examples of quadratic equations: y = x2 y = x2 + 1 y = x2 + 3x - 10

Quadratic Equations These equations can be drawn as graphs in the same way that we know how to draw straight line graphs. By substituting the given values for x calculate the y value for the following equation: y = x2 x -3 -2 -1 1 2 3 y   9 4 1 1 4 9 This generates coordinates to plot on a graph (-3,9) (1, 1) Notice how the y values are symmetrical (-2,4) (2, 4) (-1, 1) (3, 9) (0, 0)

The lowest point is called Plotting these points Quadratic Equations (-3,9) (1, 1) (-2,4) (2, 4) 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y (-1, 1) (3, 9) x x (0, 0) What is the name of this shape? x x Parabola x x Why is it this shape? x because x2 is always positive, even when x itself is negative The lowest point is called the ‘minimum’ and the value of y increases at a much faster rate than x

Draw the graph for the following quadratic function y = x2 + 1 x x Quadratic Equations Draw the graph for the following quadratic function 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y y = x2 + 1 x x Complete the table: x -3 -1 1 3 y   10 2 1 2 10 x x Draw the graph x

Quadratic Equations Drawing graphs of more complex quadratic functions we need bigger tables with more data in order to be able to draw the graphs accurately Draw the graph of this function y = x2 - 3x - 2 x -4 -3 -2 -1 1 2 3 4 x2   -3x y 16 9 4 1 0 1 4 9 16 12 9 6 3 0 -3 -6 -9 -12 Now add all the values together to find y -2 -2 -2 -2 -2 -2 -2 -2 -2 26 16 8 2 -2 -4 -4 -2 2

Quadratic Equations y x 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 1 2 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -10 x y

Draw the graph of this function y = x2 + 3 x -4 -3 -2 -1 1 2 3 4 x2 +3 Quadratic Equations x Draw the graph of this function -4 -3 -2 -1 0 1 2 3 4 20 18 16 14 2 12 10 8 6 4 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 y = x2 + 3 x -4 -3 -2 -1 1 2 3 4 x2   +3 y 16 9 4 1 3 3 3 3 3 3 3 3 3 19 12 7 4 3 4 7 12 19

Draw the graph of this function x Quadratic Equations Draw the graph of this function x -4 -3 -2 -1 0 1 2 3 4 10 9 8 7 2 6 5 4 3 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = x2 + 2x + 1 x -4 -3 -2 -1 1 2 3 4 x2 2x +1 y   16 9 4 1 0 1 4 9 16 -8 -6 -4 -2 0 2 4 6 8 +1+1 +1+1 +1 +1+1 +1+1 9 4 1 0 1 4 9 16 25

Draw the graph of this function y = 2x2 – x + 3 x -4 -3 -2 -1 1 2 3 4 Quadratic Equations Draw the graph of this function -4 -3 -2 -1 0 1 2 3 4 50 45 40 35 2 30 25 20 15 10 5 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 X y = 2x2 – x + 3 x -4 -3 -2 -1 1 2 3 4 2x2   -x +3 y 32 18 8 2 0 2 8 18 32 +4+3 +2+1 0 -1 -2 -3 -4 +3+3 +3+3 +3+3+3+3+3 37 2413 6 3 4 9 18 31

Draw the graph of this function y = x2 + 3 x -4 -3 -2 -1 1 2 3 4 x2 +3 Quadratic Equations Worksheet 1 Draw the graph of this function -4 -3 -2 -1 0 1 2 3 4 20 18 16 14 2 12 10 8 6 4 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 y = x2 + 3 x -4 -3 -2 -1 1 2 3 4 x2   +3 y

Draw the graph of this function y = x2 + 2x + 1 x -4 -3 -2 -1 1 2 3 4 Quadratic Equations Worksheet 2 Draw the graph of this function -4 -3 -2 -1 0 1 2 3 4 20 18 16 14 2 12 10 8 6 4 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 y = x2 + 2x + 1 x -4 -3 -2 -1 1 2 3 4 x2   2x +1 y

Draw the graph of this function y = 2x2 – x + 3 x -4 -3 -2 -1 1 2 3 4 Quadratic Equations Worksheet 3 Draw the graph of this function -4 -3 -2 -1 0 1 2 3 4 10 9 8 7 2 6 5 4 3 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 y = 2x2 – x + 3 x -4 -3 -2 -1 1 2 3 4   y