Sin, Cos, Tan
Linear functions can be written in the form y = mx + c where y and x are variables and m and c are constants (numbers).
If you write them like this then m is the gradient and c is the y-intercept (point where it crosses the y-axis). The graphs of linear functions are straight lines.
To find m:
Pick any two points.
To find c:
c is the point where the graph crosses the y-axis.
Quadratic functions can be written in the form:
y = ax2 + bx + c
where a, b and c are constants and 'a' doesn't equal zero.
Quadratic graphs are always parabolas ('U' shapes).
The really important bits of a quadratic are:
Where it turns (the bottom of the 'U')
Where it crosses the x-axis (if it does!)
The solutions of a quadratic are where the graph crosses the x-axis!
Cubics and Reciprocals
You need to be able to:
- Plot and draw these.
- Recognise the shapes.
- Read the solutions from the graph (cubics only).
Cubics can be written in the form:
y = ax3 + bx2 + cx + d
Reciprocals are where the x is on the bottom of a fraction.
Drawing their graphs - Table - Axes - Plot - Draw - Label
The solutions of a cubic are where it crosses the x-axis and it can have up to 3.
Graphs of simultaneous equations
As simultaneous equations at GCSE are linear (can both eb written in the form y = mx + c) their graphs will be straight lines.
The solution (x-value and y-value) is where the straight lines intersect (cross one another).
Inequalities - regions on a graph
To draw a graph:
- Change the inequality sign to an '=' sign.
- By choosing 4 or 5 different values for x, make a table of co-ordinates.
- Draw and label the line (make it dotted if the inequality sign is < or >).
- Choose a test point (not on the line!).
- Put the x and y values of the test point into the inequality.
- If it works, shade and label that side of the line with the inequality.
- If it doesn't work, shade and label the other side.
If you show a graph of a journey showing distance travelled (on the y-axis) against time (on the x-axis):
- The gradient (or slope) of the graph represents the speed.
- A horizontal section indicates that you have stopped.
- A section sloping up means that you are going away.
- A section sloping down means you are coming back.
- The steeper the line, the faster you are going.
- The gradient (or slope) of the graph represents the acceleration.
- The area under the graph (for any section) is the distance travelled (in that section).
- A horizontal section indicates constant speed (no acceleration).
- A section sloping up means accelerating.
- A section sloping down means slowing down.
- The steeper the line, the quicker the acceleration.
1cm = 10mm
1m = 100cm
1km = 1000m
1kg = 1000g
1 tonne = 1000kg
1 litre = 1000ml = 1000cm3
Note to remember:
1m = 100cm 1m2 = 10 000cm2 1m3 = 1 000 000cm3
Here are some more facts that you should know:
1 foot = 12 inches
1 yard = 3 feet
1 mile = 1760 yards
1 pound = 16 ounces
1 stone = 14 pounds
1 ton = 160 stones (or 2240 pounds)
Note: the different spellings of tonne (metric) and ton (imperial).
Volume: 1 gallon = 8 pints
Kilometres and Miles:
Miles to Km - Multiply by 1.6
Km to Miles - Multiply by 0.62
Kilograms and Pounds:
Kg to Pounds - Multiply by 2.2
Pounds to Kg - Multiply by 0.45
Litres and Gallons:
Litres to Gallons - Multiply by 0.22
Gallons to Litres - Multiply by 4.55
Metres, centimetres, feet and inches:
Inch to Cm - Multiply by 2.54
Cm to Inch - Multiply by 0.39
A locus is simply a set of points that satisfy some sort of condition.
Distance from a point
A circle around the point!
Distance from two points
A perpendicular line straight down the middle of the points:
- Set your compasses so that they are roughly the same as the distance between the points (or less if you don't have a lot of room!).
- Put the point of the compasses on the first cross and do two arcs - one above the points and one below.
- Put the point on the second cross and do the same thing so that you cross the first arcs (making sure you keep the compasses the same distance apart).
- Now simply draw a line straight down the middle through the points where the arcs cross.
Distance from two lines
The set of points that are the same distance from two lines is a straight line down the middle which bisects the angle (cuts it in half):
- Get a pair of compasses and place the point where the two lines meet. Draw little arcs that cross each of the lines.
- Now, keeping the compasses set, put the point on each line where your arc has crossed it and draw another little arc in-between the two lines. You should now have another two little arcs in the middle.
- Draw a straight line from the angle through the point where your little arcs cross and you've done it!