Linear graphs

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 graphs

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!

Cubic and reciprocal graphs

You need to be able to:

1. Plot and draw these.
2. Recognise the shapes.
3. 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:

1. Change the inequality sign to an '=' sign.
2. By choosing 4 or 5 different values for x, make a table of co-ordinates.
3. Draw and label the line (make it dotted if the inequality sign is < or >).
4. Choose a test point (not on the line!).
5. Put the x and y values of the test point into the inequality.
6. If it works, shade and label that side of the line with the inequality.
7. If it doesn't work, shade and label the other side.

Travel graphs

Distance/time

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.

Speed/time

• 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.