MISSION · Algebra
Straight-Line Graphs and y = mx + b
Two numbers are enough to draw any straight line
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LEVEL 2 · LEARN
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The idea in one sentence
Every straight line is described by where it starts and how fast it climbs, and those are exactly the two numbers in \(y = mx + b\).
The two numbers
\(m\) is the slope — the climb per step right. \(b\) is the y-intercept — the value of \(y\) when \(x = 0\), which is where the line crosses the vertical axis.
Give me those two and I can draw your line without a table of values. Put a dot at \((0, b)\), then step one right and \(m\) up, over and over. That’s the whole procedure.
\(y = 2x + 3\) starts at \((0, 3)\) and climbs 2 for every 1 across.
Why it’s a straight line at all
Because the slope never changes. A constant rate of change, stretched across the whole number line, can only draw something straight — any bend would mean the rate changed somewhere.
That’s also how you spot a linear relationship in a table. Look at the \(y\) values as \(x\) goes up in equal steps. If they change by the same amount each time, the relationship is linear and that amount is \(m\).
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(y\) | 3 | 5 | 7 | 9 |
Up 2 every time, starting from 3. So \(y = 2x + 3\), read straight off the table.
The story version
The form is worth recognising in words, because it turns up constantly outside maths class:
- A gym costs £20 to join and £15 a month: \(C = 15m + 20\).
- A tank holds 500 litres and drains 8 a minute: \(V = -8t + 500\).
\(b\) is the starting amount, the part you owe before anything happens. \(m\) is the rate, the part that accumulates. A negative \(m\) just means the quantity is going down — the line falls, and the story is about draining rather than filling.
Two special cases
A slope of 0 gives \(y = b\), a flat horizontal line at height \(b\). Nothing changes as \(x\) moves.
A vertical line can’t be written this way at all. It has no slope to put in \(m\), so it gets its own form, \(x = 4\), meaning every point whose \(x\) is 4 regardless of \(y\). Vertical lines are not functions, because one \(x\) would have to give infinitely many \(y\) values.
Common mix-ups
- Swapping \(m\) and \(b\). The one attached to \(x\) is the slope.
- Plotting the intercept on the wrong axis. \(b\) is on the \(y\)-axis, found by setting \(x = 0\).
- Reading a rearranged equation too fast. In \(y = 7 - 2x\) the slope is \(-2\), not
- Sort it into \(mx + b\) order before reading it.
- Assuming every equation with \(x\) and \(y\) is a line. \(y = x^2\) is a curve.
Check yourself
Where does \(y = -3x + 6\) cross the \(y\)-axis, and what happens to \(y\) each time \(x\) goes up by one? Where does it cross the \(x\)-axis?
It crosses the \(y\)-axis at \((0, 6)\), and \(y\) drops by 3 for each step right. It meets the \(x\)-axis where \(y = 0\): \(0 = -3x + 6\) gives \(x = 2\).
LEVEL 3 · REFLECT
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