Quadratic Graph Drawing.
y = 4x2 – 8x – 5
y = 4x2 – 4x – 3
y
-0.5
-5
-9
y
1
2 .5 x
-1/2
-3
3/2
x
Quadratic Graph Structure.
Consider the typical quadratic graph shape below:
y
2
x
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:
(1) Point of intersection with the
y axis.
y
“Graph cuts y axis when x = 0”
2
(2) Point of intersection with x
axis.
x “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.
y
Graph cuts y axis when x = 0
 y  0  9  9
Point ( 0, -9 ) is on the graph.
-3
3
x
(2) Find the x axis intercept.
Graph cuts x axis when y = 0.
-9
 x2  9  0
(x – 3 ) (x + 3 ) = 0
x=3
x=-3
The points (3,0) and (-3,0) are on the graph.
(3) Turning point of graph.
y
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.
-3
-9
3
x
Example 2
Sketch the graph of y = x 2 - 25
y
(1) Find the y axis intercept.
Graph cuts y axis when x = 0
 y  0  25  25
Point ( 0, -25 ) is on the graph.
-5
(2) Find the x axis intercept.
-25
5
x
Graph cuts x axis when y = 0.
 x 2  25  0
(x – 5 ) (x + 5 ) = 0
x=5
x=-5
The points (5,0) and (-5,0) are on the graph.
(3) Turning point of graph.
y
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).
-5
Now sketch the graph.
-25
5
x
What Goes In The Box ? 1
Sketch the graph of the following functions:
(1) y = x2 - 4
y
-2
-4
(2) y = x2 - 36
y
Y = x2 - 4
2
x
-6
-36
Y = x2 - 36
6
x
Graphs Of Type y =
Sketch the graph of y = x 2 + 2x –24
2
ax
+ bx + c
y
(1) Find the y axis intercept.
Graph cuts y axis when x = 0
-6
4
x
 y  0  0  24  24
The point (0,-24) is on the graph.
-24
(2) Find the x axis intercept.
Graph cuts x axis when y = 0.
 x 2  2 x  24  0
(x + 6) (x – 4) = 0
x=-6
x=4
The points (-6,0) and (4,0) are on the graph.
(3) Turning point of graph.
y = x 2 + 2x –24
Turning point x coordinate is the
midpoint of the roots.
x
64 2

 1
2
2
y
-6
-1
4
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.
-24
-25
x
Example 2.
Sketch the graph of y = 4x2 – 4x – 3
y
(1) Find the y axis intercept.
Graph cuts y axis when x = 0
-1/2
3/2
Y= –3
The point (0,-3) is on the graph.
(2) Find the x axis intercept.
-3
Graph cuts x axis when y = 0.
4x2  4x  3  0
( 2x + 1 )( 2x – 3 ) = 0
1
x
2
x
3
2
The points (-1/2,0) and (3/2,0) are on the graph.
x
(3) Turning point of graph.
y = 4x2 – 4x – 3
Turning point x coordinate is the
midpoint of the roots.
1 3

1
2
2
x

2
2
y
-1/2
1/2
3/2
Substitute x = ½ into the equation
y = 4x2 – 4x – 3 to find the y coordinate.
2
1
1
y  4  4  3
2
2
-3
-4
y= - 4
Minimum turning point (1/2, -4)
Now sketch the graph.
x
What Goes In The Box? 2
Sketch the graphs of the equations given below:
(1)
(2) y = 4x2 – 8x – 5
y = x2 – 2x – 8
y
y
-2
-8
-9
1
4 x
-0.5
-5
-9
1
2 .5 x
Graph Of Type y = a2 – x2
Sketch the graph of y = 25 – x2
y
25
(1) Find the y axis intercept.
y = 25 – x2
Graph cuts y axis when x = 0
y = 25
The point ( 0,25 ) is on the graph
(2) Find the x axis intercept.
Graph cuts x axis when y = 0.
-5
5
x
25 – x2 = 0
(5– x)(5+x)=0
x=5
x=-5
The points (0,5 and (0,-5) are on the graph.
Now sketch the graph.
The effect of the
negative x2 is to
make the graph “A”
shaped.
Example 2
Sketch the graph of y = – x2 + x + 6
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) Find the x axis intercept.
-2
3
Graph cuts x axis when y = 0.
– x2 + x + 6 =0
Divide throughout by -1
x2 - x - 6 =0
( x – 3) ( x + 2 ) = 0
x=3
x=-2
The points ( 3, 0 ) and ( -2 , 0 ) are on the graph.
x
(3) Turning point of graph.
Turning point x coordinate is the
midpoint of the roots.
23 1
x

2
2
y = - x2 + x + 6
y
6.25
6
Substitute x = ½ into the equation
y = - x2 + x + 6 to find the y coordinate.
2
1
1
1
y  ( )   6  6
2
2
4
The maximum turning point is ( 0.5, 6.25 )
Now sketch the graph.
-2
0.5
3
x
What Goes In The Box ? 3
Sketch graphs of the quadratic equations below:
(1)
y = 36 – x2
(2)
y = - x2 + 2x + 8
y
y
36
-6
9
8
6
x
-2
1
4
x