Transcript GCSE Revision 101

© Daniel Holloway
Any quadratic graph has x2 in its equation
We can work out how to plot a quadratic
graph using an x and y values table
If we draw out the x and y table for the quadratic equation y = x2 we should
get something like this:
x -5 -4 -3 -2 -1 0
1
2
3
4
5
x2
25
16
9
4
1
0
1
4
9
16
25
y
25
16
9
4
1
0
1
4
9
16
25
y
Next, we plot the points…
35
30
We know that when x = -5, y = 25
25
We know that when x = -4, y = 16
20
And so on for the other values
15
Finally, we connect the points with
a smooth curve
10
5
-5
0
5 x
There may be more complicated graphs to
plot involving quadratics
Take the graph for y = x2 + 5x + 6
x
-5
-4
-3
-2
-1
0
1
2
3
x2
25
16
9
4
1
0
1
4
9
5x
-25
-20
-15
-10
-5
0
5
10
15
6
6
6
6
6
6
6
6
6
6
y
6
2
0
0
2
6
12
20
30
x
-5
-4
-3
-2
-1
0
1
2
3
y
6
2
0
0
2
6
12
20
30
y
35
30
25
20
15
10
5
-5
0
5 x
We can use graphs with quadratics in them
to solve quadratic equations
When we draw quadratic lines on a graph,
it crosses the x-axis at two points. Since the
x-axis is the line y = 0, any point along in has
a y value of zero
We call the “answers” to the equation its
roots
Take the graph for y = 2x2 - 5x - 3 for -2 ≤ x ≤ 4
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x2
4
2.25
1
0.25
0
0.25
1
2.25
4
6.25
9
12.25
16
2x2
8
4.5
2
0.5
0
0.5
2
4.5
8
12.5
18
24.5
32
5x
-10
-7.5
-5
-2.5
0
2.5
5
7.5
10
12.5
15
17.5
20
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
y
15
9
4
0
-3
-5
-6
-6
-5
-3
0
4
9
We could plot it and then look at the points
at which the line crosses the x-axis
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
y
15
9
4
0
-3
-5
-6
-6
-5
-3
0
4
9
y
With the graph complete, we
can easily spot the two points
where it crosses the x-axis
(although with this graph you
could tell these points by looking
at the table, usually you will
need to draw the graph as they
are not integers)
15
10
5
-2
-1
O
-5
-10
1
x
The points
2
3 are
4 x = -0.5 and x = 3.5
where y = 0. So we have solved
the equation 0 = 2x2 – 5x – 3 which
is 2x2 – 5x – 3 = 0
Because squaring a negative number gives
a positive result, there is only one pair of
coordinates on a y = x2 graph for each x
value. However, the coordinates of y = √x
come in two pairs:
 when x = 1, y = ±1 giving two coordinates:
(1,-1) and (1,1)
 when x = 4, y = ±2 giving two coordinates:
(4,-2) and (4,2)
y
2
1
-1
O
-1
-2
1
2
3
4
5
x
We can use those
points to plot the
graph y = √x
 x = 0, y = 0
 x = 1, y = 1
 x = 1, y = -1
 x = 2, y = 2
 x = 2, y = -2
A reciprocal equation takes the form:
a
y= x
All reciprocal graphs have a similar shape
and certain symmetrical properties
1
Take the graph for y = x
y
5
-4
-2
O
-5
2
4
x
x
y
-0.8
-4
-0.25
-1.25
-0.6
-3
-0.33
-1.67
-0.4
-2
-0.5
-2.5
-0.2
-1
-1
-5
0.2
1
1
5
0.4
2
0.5
2.5
0.6
3
0.33
1.67
0.8
4
0.25
1.25
Note
is no
valueas
forit x = 0
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