Drawing linear graphs.
Turn a linear equation into points, then join them with a straight line.
Build tables · plot straight lines · use intercepts · recognise horizontal and vertical lines
Two points define a line; a third checks it
A linear equation describes many pairs of coordinates, not just one answer. For y = 2x + 1, every chosen input x produces a matching output y, and every matching pair lies on the same straight line.
Choose convenient x-values, substitute them into the equation, and record the resulting pairs in a table. Plot at least three to help catch arithmetic errors, then use a ruler for the straight line. Two correct points determine the line; the third acts as a check.
Choose convenient x-values, calculate y, plot the coordinate pairs and use a ruler to draw the straight line through them. Extend it across the required graphing interval.
| x | 0 | 1 | 2 |
|---|---|---|---|
| y | 5 | 3 | 1 |
Draw y = −2x + 5.
At x = 0, y = 5. At x = 1, y = 3. At x = 2, y = 1. Plot (0, 5), (1, 3), (2, 1). All three lie on the same straight line.
Pause and explainFor y = 2x + 1, what point corresponds to x = 0?
(0, 1), because the equation gives y = 1.
Rearrange or use intercepts
An equation may describe a straight line even when y is not isolated. Rearranging it makes substitution easier, but does not change the pairs that satisfy the equation.
Intercepts give another route to plotting. At the y-axis, x = 0; at the x-axis, y = 0. Substitute those zero values separately, find the two crossing points, and draw the line through them. Use another method if both intercepts are the same origin point.
Draw 3x + 2y = 5.
y = −32 x + 52
Alternatively set x = 0 to get the y-intercept (0, 2.5), and y = 0 to get the x-intercept (53, 0). A point such as (1, 1) provides a useful check.
Horizontal: y = k
Every point has the same y-coordinate. For y = 3, plot (−2, 3) and (4, 3).
Vertical: x = k
Every point has the same x-coordinate. For x = −2, plot (−2, −3) and (−2, 4).
Pause and explainWhy set y = 0 to find the x-intercept?
Every point on the x-axis has zero vertical coordinate, so the crossing must satisfy y = 0.
Your turn. Check with substitution.
First decide whether a value table or intercepts will produce convenient plotting points. Show the substitutions so an arithmetic error can be traced.
Draw with a ruler, label the equation and extend the line through the required grid region. Check a plotted point in the original equation, especially after rearranging.
Give three points for y = 7 − 4x using x = 0, 1, 2.
Reveal worked answer
Find the intercepts of 2x + 3y = 6.
Reveal worked answer
Describe the graphs x = 4 and y = −2.