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Mathematics and Statistics Help CentreSwinburne University of Technology, Melbourne, Australia

GRAPHS AND EQUATIONS OF SOME STANDARD CURVES

SOME KEY POINTS IN CO-ORDINATE GEOMETRY

1. The co-ordinate axes are the lines x = 0 (the y-axis) and y = 0 (the x -axis).

2. The co-ordinates of a point are made up of: the abscissa or x -ordinate and the ordinateor y -ordinate. The co-ordinates are given as the ordered pair: ( x, y)

4. The x -intercept is the point where the graph cuts the x-axis and is found by substituting y = 0 in the equation of the graph. The y -intercept is the point where the graph cuts the y -axis and is found by substituting x = 0 in the equation of the graph.

5. The distance between two points, P( x 1, y 1) and Q( x 2, y 2) is found using Pythagoras'

Theorem: D x2 x1 2 y2 y1 2

6. The slope or gradient of a line is given by y x

y2 y1 x2 x1

If two lines are parallel, then their gradients are equal. If two lines are perpendicular, thenthe product of their gradients is equal to 1

7. The midpoint of a straight line is found by averaging the x and y ordinates of the end- points of the line:

The mid-point is x1 x2

2, y1 y2

2

8. The general equation of a straight line is given by y = m x + c m is the gradient of the line and c is the y -intercept.

9. The general equation of a circle is given by( x h)2 + ( y k )2 = r 2

The point ( h, k ) is the centre of the circleand the radius is r units in length.

A circle can be graphed using thegraphics calculator by rewriting theequation as two functions of y:

y r 2 x h 2 k

For example the circle

y 4 x 1 2 3has centre at (1, 3) and radius 2

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Mathematics and Statistics Help CentreSwinburne University of Technology, Melbourne, Australia

12 The general equation of a parabola is of the form y = ax 2 + bx + c

The magnitude of a affects the spread of the graph: if a is large, then the graph is 'narrow',if a is small, then the graph is 'wide'.

The sign of a affects the nature of the turning point.If a is positive, then the parabola is concave upwards and the graph has a minimum value.If a is negative, then the parabola is concave downwards and the graph has a maximumvalue.

The x-intercepts can be found by substituting y = 0 in the original equation.

i.e. ax 2 + bx + c = 0. This gives xb b 2 4ac

2a

The y-intercept is given by c

The turning point can be found by calculus.The turning point of the graph is more appropriately called the vertex of the parabola.The axis of symmetry goes through the vertex of the parabola.The maximum / minimum value of the function corresponds to the value of y at thevertex.

If the equation is written in the form y a xb

2a

2

cb 2

4aby completing the square,

then the coordinates of the vertex areb

2a,c

b 2

4a

Other forms of parabolas may have their equation in the form x = ay 2 + by + c If a is positive, the parabola opens to the right: if a is negative, then the parabolaopens to the left.

Some examples: y = 2 x2 + 4 x 6 = 2( x + 1) 2 8 x = 3 y2 + 12 y 11 = 3( y 2) 2 + 1

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Mathematics and Statistics Help CentreSwinburne University of Technology, Melbourne, Australia

7/28/2019 Standard Curves

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Mathematics and Statistics Help CentreSwinburne University of Technology, Melbourne, Australia