Quadratic Graph Drawing. y = 4x2 – 8x – 5 y = 4x2 – 4x – 3 x y -0.5 2 .5 1 -9 -5 x y -1/2 3/2 -3

Quadratic Graph Structure. Consider the typical quadratic graph shape below: x y 2 1 3 To draw a quadratic graph you must be able to calculate all the points numbered on the diagram:

The significance of each point and the starting point to finding the point are given below: Point of intersection with the y axis. x y “Graph cuts y axis when x = 0” 2 (2) Point of intersection with x axis. “Graph cuts x axis when y=0” 1 3 These are “the roots” of the equation. (3) Turning point of graph. Turning point x coordinate is the midpoint of the roots.

Graph Of Type y = x2 – a2 Sketch the graph of y = x 2 - 9 (1) Find the y axis intercept. -9 x y Graph cuts y axis when x = 0 -3 3 Point ( 0, -9 ) is on the graph. (2) Find the x axis intercept. Graph cuts x axis when y = 0. (x – 3 ) (x + 3 ) = 0 The points (3,0) and (-3,0) are on the graph. x = 3 x = - 3

(3) Turning point of graph. -9 x y -3 3 Y = x2 - 9 Turning point x coordinate is the midpoint of the roots. From the symmetry of the graph we can see that this point must be (0,-9). Now sketch the graph.

Example 2 Sketch the graph of y = x 2 - 25 -25 x y (1) Find the y axis intercept. Graph cuts y axis when x = 0 -5 5 Point ( 0, -25 ) is on the graph. (2) Find the x axis intercept. Graph cuts x axis when y = 0. (x – 5 ) (x + 5 ) = 0 The points (5,0) and (-5,0) are on the graph. x = 5 x = - 5

(3) Turning point of graph. -25 x y -5 5 Y = x2 - 25 Turning point x coordinate is the midpoint of the roots. From the symmetry of the graph we can see that this point must be (0,-25). Now sketch the graph.

What Goes In The Box ? 1 Sketch the graph of the following functions: (1) y = x2 - 4 (2) y = x2 - 36 -4 x y -2 2 Y = x2 - 4 -36 x y -6 6 Y = x2 - 36

Graphs Of Type y = ax2 + bx + c Sketch the graph of y = x 2 + 2x –24 x y (1) Find the y axis intercept. -6 4 Graph cuts y axis when x = 0 The point (0,-24) is on the graph. -24 (2) Find the x axis intercept. Graph cuts x axis when y = 0. (x + 6) (x – 4) = 0 The points (-6,0) and (4,0) are on the graph. x = - 6 x = 4

(3) Turning point of graph. y = x 2 + 2x –24 Turning point x coordinate is the midpoint of the roots. x y -6 4 -24 -25 -1 Substitute x = -1 into the equation y = x 2 + 2x –24 to find the y coordinate. y = (-1)2 +(2 x –1) – 24 = - 25 The point (-1,-25) is the minimum turning point. Now sketch the graph.

Example 2. Sketch the graph of y = 4x2 – 4x – 3 x y (1) Find the y axis intercept. -1/2 3/2 Graph cuts y axis when x = 0 Y = – 3 The point (0,-3) is on the graph. (2) Find the x axis intercept. -3 Graph cuts x axis when y = 0. ( 2x + 1 )( 2x – 3 ) = 0 The points (-1/2,0) and (3/2,0) are on the graph.

(3) Turning point of graph. y = 4x2 – 4x – 3 Turning point x coordinate is the midpoint of the roots. x y -1/2 3/2 -3 1/2 -4 Substitute x = ½ into the equation y = 4x2 – 4x – 3 to find the y coordinate. y= - 4 Minimum turning point (1/2, -4) Now sketch the graph.

What Goes In The Box? 2 Sketch the graphs of the equations given below: (2) y = 4x2 – 8x – 5 y = x2 – 2x – 8 (1) x y -0.5 2 .5 1 -9 -5 x y -2 4 1 -8 -9

Graph Of Type y = a2 – x2 Sketch the graph of y = 25 – x2 x y (1) Find the y axis intercept. Graph cuts y axis when x = 0 y = 25 The point ( 0,25 ) is on the graph (2) Find the x axis intercept. 5 -5 Graph cuts x axis when y = 0. 25 – x2 = 0 ( 5 – x ) ( 5 + x ) = 0 The effect of the negative x2 is to make the graph “A” shaped. x = 5 x = - 5 The points (0,5 and (0,-5) are on the graph. Now sketch the graph.

Example 2 Sketch the graph of y = – x2 + x + 6 x y 6 (1) Find the y axis intercept. Graph cuts y axis when x = 0 y = 6 The point ( 0,6 ) is on the graph -2 3 (2) Find the x axis intercept. Graph cuts x axis when y = 0. – x2 + x + 6 =0 Divide throughout by -1 x2 - x - 6 =0 ( x – 3) ( x + 2 ) = 0 The points ( 3, 0 ) and ( -2 , 0 ) are on the graph. x = 3 x = - 2

(3) Turning point of graph. y = - x2 + x + 6 Turning point x coordinate is the midpoint of the roots. x y 6 -2 3 0.5 6.25 Substitute x = ½ into the equation y = - x2 + x + 6 to find the y coordinate. The maximum turning point is ( 0.5, 6.25 ) Now sketch the graph.

What Goes In The Box ? 3 Sketch graphs of the quadratic equations below: (1) y = 36 – x2 (2) y = - x2 + 2x + 8 1 x y 8 -2 4 9 x y 6 -6 36