Reading & Plotting Linear Graphs
Learn to plot and read linear graphs for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-01). A rule such as \(y = 2x + 1\) gives a straight line when it is graphed, and reading that line well is the first step towards solving simultaneous equations on a graph.
You will fill in a table of values, plot the points and rule a line through them, read values off the line, test whether a point sits on it, and find the \(y\)-intercept and the gradient — skills for reading costs, savings and other everyday straight-line graphs.
Theory
A linear rule such as \(y=mx+c\) graphs as a straight line. This Year 12 Standard 1 (NSW) guide shows how to build a table of values, plot and read the line, check whether a point lies on it, and find the intercepts and the gradient.
A linear rule such as \(y=2x-2\) gives a straight line when it is graphed. To plot it, make a table of values: substitute each \(x\)-value into the rule to find \(y\), plot each point \((x,y)\), then join the points with a ruler.
A point is written \((x,y)\): go across to \(x\) first, then up (or down) to \(y\). A point lies on the line if substituting its \(x\)-value into the rule gives its \(y\)-value.
In \(y=mx+c\), \(c\) is the \(y\)-intercept (where \(x=0\)) and \(m\) is the gradient, the rise over the run. A positive gradient rises to the right; a negative gradient falls. The \(x\)-intercept is where the line crosses the \(x\)-axis, so \(y=0\) there.
A linear rule written in gradient–intercept form:
The gradient from two points on the line:
A table of values for \(y=2x-2\):
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(y=2x-2\) | \(-2\) | 0 | 2 | 4 | 6 |
How to plot and read a linear graph
- Substitute each \(x\)-value into the rule (use brackets with negatives) to complete a table of values.
- Plot each point \((x,y)\) — across first, then up or down — and join the points with a ruler.
- Read values from the line: up from \(x\) to the line, then across to the \(y\)-axis (or the reverse to find \(x\)).
- Find the intercepts (\(x=0\) and \(y=0\)) and the gradient \(\dfrac{\text{rise}}{\text{run}}\).
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(y\) | ? | ? | ? | ? |
Substitute each \(x\)-value into \(y=4x-3\).
| \(x=0:\ y\) | \(=\) | \(4(0)-3=-3\) |
| \(x=1:\ y\) | \(=\) | \(4(1)-3=1\) |
| \(x=2:\ y\) | \(=\) | \(4(2)-3=5\) |
| \(x=3:\ y\) | \(=\) | \(4(3)-3=9\) |
The \(y\)-values are \(-3,\ 1,\ 5,\ 9\), and the point is \((2,5)\).
Substitute the \(x\)-value and compare with the \(y\)-value.
| \(\text{(a) } y\) | \(=\) | \(2(3)+5=11\ \checkmark\) |
| \(\text{(b) } y\) | \(=\) | \(2(-1)+5=3\) |
| \(3\) | \(\ne\) | \(2\) |
\((3,11)\) lies on the line; \((-1,2)\) does not.
(a) Go up from \(x=1\) to the line, then across: \(y=4\). (b) Use \(x=0\), then \(y=0\).
| \(y\) | \(=\) | \(-2(1)+6=4\ \checkmark\) |
| \(x=0:\ y\) | \(=\) | \(-2(0)+6=6\) |
| \(0\) | \(=\) | \(-2x+6\) |
| \(2x\) | \(=\) | \(6\) |
| \(x\) | \(=\) | \(3\) |
\(y=\mathbf{4}\); \(y\)-intercept 6, \(x\)-intercept 3.
Gradient \(=\dfrac{\text{rise}}{\text{run}}\) between the two points.
| \(\text{rise}\) | \(=\) | \(5-(-1)=6\) |
| \(\text{run}\) | \(=\) | \(2-0=2\) |
| \(m\) | \(=\) | \(\dfrac{6}{2}=3\) |
| \(c\) | \(=\) | \(-1\) |
Gradient 3, and the equation is \(y=3x-1\).
Common pitfalls
Frequently asked questions
How do you plot a linear graph from a table of values?
Substitute each x-value into the rule to find its y-value, plot each point with x across and y up, then join the points with a ruler. All the points of a linear rule lie on one straight line.
How do you check if a point is on a line?
Substitute the x-value of the point into the rule. If the answer equals the y-value of the point, the point lies on the line. If not, it does not.
What is the y-intercept of a line?
It is where the line crosses the y-axis, which happens when x equals 0. In the rule y equals mx plus c, the y-intercept is c.
How do you find the x-intercept?
The x-intercept is where the line crosses the x-axis, so y equals 0 there. Put y equal to 0 in the rule and solve for x, or read it straight off the graph.
How do you find the gradient of a line from a graph?
Pick two points on the line and divide the rise (the change in y) by the run (the change in x). A line rising to the right has a positive gradient; a falling line has a negative gradient.