From function to graph
A graph of a function is a picture of the function on the Cartesian plane. Every point on the graph has coordinates (x, y), where y is the output f(x). To draw the graph we build a table of values: choose several x-values, work out each y, then plot the points and join them.
Key idea
A graph like y = ax + b (highest power of x is 1) is a straight line. A graph like y = x² (x is squared) is a smooth curve called a parabola.
Making a table of values
Worked example
Draw y = 2x + 1 for x from 0 to 3.
| x | 0 | 1 | 2 | 3 |
| y | 1 | 3 | 5 | 7 |
For x = 2: y = 2(2) + 1 = 5. Plot (0,1), (1,3), (2,5), (3,7) and join them with a ruler — a straight line.
Reading a graph
To find y for a given x, go up from that x-value to the line, then across to the y-axis. To find x for a given y, go across from the y-value to the line, then down to the x-axis. Always check that a chosen point makes the equation true, for example (2, 7) satisfies y = 2x + 1 because 2(2) + 1 = 5 — so (2, 7) is not on that line.
Remember
- Substitute each x to get y before plotting.
- Linear function → straight line; quadratic (x²) → curve.
- A point lies on the graph only if it satisfies the equation.