1. 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

2. 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:

3. 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.

4. 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

5. (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.

6. Example 2
Sketch the graph of y = x
-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

7. (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.

8. 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

9. 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

10. (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.

11. 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.

12. (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.

13. 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

14. 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.

15. 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

16. (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.

17. 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