Contents:
Simple Equations: These
can be plotted on a graph by finding the value of 'y' when 'x' = 0, and finding
the value of 'x' when 'y' is zero. 
Example: Draw a graph for: 6x + 2y =18.
When x = 0, 2y = 18, therefore y = 9.
When y = 0, 6x = 18, therefore x = 3.
The line is drawn through (0, 9) and (3, 0).
Rearranging 6x + 2y = 18 to (y = mx + c) =
y = -3x + 9.
The equation may be found from the graph:
Gradient is (Y 0-9 = X 0-3) downhill = -3,
-3x.
The line crosses the 'y' axis at 9.
Hence the equation y = -3x + 9.
The Quadratic: Table of Values for x
and x:
A graph of a quadratic equation may be plotted with the help of a table. Calculate
for a few different values of x for y, and use the results to draw the graph.
Example: Complete a table of values for:
y = x + 2x - 8.
ie. for (x = -6). y = -6
-12 - 8. y = 36 - 20. y = 16.
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-6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
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36 | 25 | 16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 | 16 |
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-12 | -10 | -8 | -6 | -4 | -2 | 0 | 2 | 4 | 6 | 8 |
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-8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 | -8 |
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16 | 7 | 0 | -5 | -8 | -9 | -8 | -5 | 0 | 7 | 16 |
Sketch the graph:
The equation can be solved simply by checking where the line passes over the 'x' axis, ie when x is zero.
The roots are: -4 and 2.
The equation can be found from the graph:
The shape of the graph = quadratic, x.
The line crosses the 'x' axis at +2x.
The line crosses the 'y' axis at -8.
Hence the equation x+
2x - 8.
As seen in the diagram above the quadratic function has a parabola shape, or inverted parabola for a negative or -x. The roots are where the line crosses the 'x' axis. If the line just touches the 'x' axis then it has two equal roots at the value of the point. If the line does not touch the 'x' axis at all the equation is found by using complex numbers.
The Cubic: Table of values for x
and xand x: A graph of
a cubic equation may be plotted with the help of a table. Calculate for a few
different values of x for y, and use the results to draw the graph.
Example: Complete a table of values for:
y = x- 4x
- 2x + 3.
ie. for (x = -6). y = -6
-12 - 8. y = 36 - 20. y = 16.
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-8
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-1
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0
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1
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8
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27
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64
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125
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-16
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-4
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0
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-4
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-16
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-36
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-64
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-100
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+4
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+2
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0
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-2
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-4
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-6
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-8
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-10
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+3
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+3
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+3
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+3
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+3
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+3
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+3
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+3
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Sketch the Graph:
The equation can be solved simply by checking where the line passes over the 'x' axis, ie when x is zero.
The roots are approx: -1 and 1 and 4
4.
The solution to the cubic equation by means of sketching a graph is much easier than by attempting algebraic method.
The cubic equation cannot be solved by the graph if the line does not cross the 'x' axis.
Intersecting Graphs: An equation mat be solved using two intersecting graphs. The x - co-ordinate of each point of intersection gives a solution (roots) to the equation.
Example: Parabola and straight line. y =
x+ 2x - 8, and y = 
x
+ 2.
(i). Sketch both the graphs on the same axis:
(ii). Write down the equation for the values of 'x' where the two graphs intersect.
At the points of intersection the values of 'y' are the same.
Therefore: y = x+
2x - 8 = x + 2.
Rearranged to form an equation: x+
x -10 = 0.
(iii). The solution of the new equation.
The value on the 'x' axis of the intersections
are approx: -19 and 88.
Simultaneous Equations: Graphical Solution: A simultaneous equation of two separate equations may be solved by plotting their straight lines onto the same graph. The co-ordinates where the lines cross is the solution for x and y. (A graphical representation of three or more simultaneous equations would require a three dimentional or more graph).
Example: y - 2x = 2.
y + x = 8.
Rearrange to standard: (y = mx + c)
y = 2x + 2.
y = -x + 8.
Sketch the Graph:
The intersection is at Y = 6 and X = 2.
Therefore: x = 2 and y = 6.
