Skill: graphing-lines

Graphing Lines & Intercepts

Turn equations into pictures and pictures into equations. Master the intercept method — plot (0, b) and the x-intercept, connect with a straightedge — plus tables of values and reading graphs back into equations.

1 The intercept method

The fastest accurate graph uses the two points a line gives you for free: the y-intercept (set x = 0) and the x-intercept (set y = 0). Plot both, lay a straightedge through them, done.

Intercepts

y-intercept: set x = 0 · x-intercept: set y = 0

From y = 2x − 4: y-intercept (0, −4); x-intercept (2, 0). Two points far apart beat two points huddled together — spread out for accuracy.

Intercepts (0, −4) and (2, 0): far apart, so the straightedge line is accurate.

Tables work too: pick easy x-values (−2, −1, 0, 1, 2), compute y, plot. And reading backwards: two marked points on a graph give you the slope, then the equation.

2 Worked examples

Intercepts, tables, and reading a graph.

Example 1 Both intercepts

Find the intercepts of y = 2x − 4.

y-intercept (x = 0): (0, −4). x-intercept (y = 0): 0 = 2x − 4, so (2, 0).

Example 2 Table of values

Complete the table for y = −x + 3 at x = 2.

y = −(2) + 3 = 1 — plot (2, 1).

Example 3 Read the slope from a graph

A graph marks (−1, 1) and (3, 3). Find the slope.

m = (3 − 1)/(3 − (−1)) = 2/4 = 1/2.

Example 4 Check a point

Is (3, 5) on y = 2x − 1?

2(3) − 1 = 5 — yes, both sides match.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Plotting two points nearly on top of each other

Wrong: (0, 1) and (0.5, 2) then freehand — the line wobbles off. Right: use the intercepts — they are far apart, so small plotting errors barely tilt the line.

2. Connecting points with a wobbly curve

Wrong: a curved connector “through” the points. Right: linear equations graph as perfectly straight lines — use a straightedge.

3. Mixing up which intercept is which

Wrong: calling (2, 0) the y-intercept. Right: the x-intercept has y = 0; the y-intercept has x = 0. The zero names the axis it sits on.

4 Quick checks

Try these yourself, then reveal the answer.

Find the intercepts of y = 3x + 6.

y-intercept (0, 6); x-intercept: 0 = 3x + 6 gives (−2, 0).

Is (4, 7) on y = 2x − 1?

2(4) − 1 = 7 — yes.

A graph marks (0, 2) and (4, 0). What is the slope?

m = (0 − 2)/(4 − 0) = −1/2.

5 Key points

Remember

Intercepts: set x = 0 for the y-intercept, y = 0 for the x-intercept.
Spread your plotted points out — intercepts are ideal.
Linear graphs are perfectly straight: always use a straightedge.
Tables: easy x-values first (−2 to 2), compute y, plot.
Reading a graph: marked points give slope, slope plus a point gives the equation.

Key vocabulary

Intercept method

Graphing by plotting the x- and y-intercepts and connecting them.

Table of values

Chosen x-values with their computed y-values, ready to plot.

Straightedge

Any rigid straight edge (ruler) for drawing truly straight lines.

Lattice point

A point with integer coordinates — the easiest to plot exactly.

Verify by substitution

Plugging a point into an equation to check it lies on the graph.

Practice graphing Open the grapher Step through a graph