📐 Grade 8 · Lesson 4 of 15
Graphing Linear Functions: Slope & Intercept
Two numbers, m and b, are all you need to draw — and understand — any straight line.
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The big idea: Every straight line can be written as y = mx + b. The slope m tells you how steep the line is and which way it tilts; the intercept b tells you where it crosses the y-axis. Change those two numbers and you can make any line you like.
A line is a rule, not just a picture
A straight line on a graph is really a rule: give it an input x, and it hands back an output y. Linear functions are the simplest rule of all — the output changes by the same amount every time the input goes up by one. That steady, predictable change is exactly what makes a line straight.
Remarkably, you only need two numbers to pin down that rule completely.
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The slope–intercept form
In y = mx + b, the number m is the slope (how much y changes each time x goes up by 1) and b is the y-intercept (the value of y when x = 0 — where the line crosses the vertical axis).
\[ y = m\,x + b \]
Slope–intercept form. m sets the steepness and direction; b sets the starting height.
Meet m and b by moving them
The fastest way to feel what m and b do is to change them and watch. Drag the two sliders below. Notice how m tilts the line (positive tilts up to the right, negative tilts down, zero is flat) and how b slides the whole line up and down without changing its steepness.
🎮 Interactive: build a line LIVE
Drag the Slope (m) and Intercept (b) sliders. The orange triangle shows the slope as rise-over-run: for every 1 you move right, the line rises by m. The dot marks where the line meets the y-axis, at (0, b).
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Why b is the crossing point
Set x = 0 in y = mx + b and the mx term vanishes, leaving y = b. So the line always passes through the point (0, b). That is why b is called the y-intercept — no calculation needed, it is just the number sitting after the x term.
Graphing a line in two moves
Once you can read m and b, graphing is quick:
- Plot the intercept. Put a dot at (0, b) on the y-axis.
- Use the slope to step. Write m as rise/run. From the intercept, move run to the right and rise up (or down, if the rise is negative), and plot a second point. Connect them.
📝 Worked example: Graph the line y = 2x − 3.
- Read off the numbers: slope m = 2, intercept b = −3.
- Plot the y-intercept first: a dot at (0, −3).
- Write the slope as rise/run: 2 = 2/1, so from (0, −3) move 1 right and 2 up to reach (1, −1).
- Draw the straight line through (0, −3) and (1, −1).
✓ A line rising steeply to the right, crossing the y-axis at −3. Check it in the tool above by setting m = 2 and b = −3.
📝 Worked example: A line crosses the y-axis at 4 and, for every 3 steps right, drops 1 step down. Write its equation.
- The y-intercept is 4, so b = 4.
- Rise over run: it drops 1 (rise = −1) over a run of 3, so m = −1/3.
- Substitute into y = mx + b.
✓ \(y = -\tfrac{1}{3}x + 4\).
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A common mix-up
The slope is the number multiplying x, not just the first number you see. In y = 5 − 2x, the slope is −2 and the intercept is 5 — reorder it as y = −2x + 5 to read them safely.
Check your understanding
1. In the equation y = −4x + 7, what is the slope?
The slope is the coefficient of x, which is −4. The 7 is the y-intercept.
2. A line has equation y = 3x − 5. Where does it cross the y-axis?
Set x = 0: y = 3(0) − 5 = −5. The line crosses the y-axis at (0, −5).
3. Which change makes a line steeper without moving where it crosses the y-axis?
Steepness is controlled by m; making |m| larger tilts the line more. Changing b slides the line up or down but keeps the same steepness.
4. A line goes through (0, 2) and rises 3 units for every 1 unit right. Its equation is…
Intercept b = 2 (it passes through (0, 2)); slope m = rise/run = 3/1 = 3. So y = 3x + 2.
✅ Key takeaways
- Any straight line can be written y = mx + b.
- m is the slope — how much y changes per step right; its sign sets the tilt.
- b is the y-intercept — the line always passes through (0, b).
- Graph a line by plotting (0, b) first, then stepping by the slope's rise over run.
- The slope is the coefficient of x — reorder the equation if needed before reading it.