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G3.15 Drawing Linear Graphs

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01 — Explanation

What do m and c tell me in y=mx+c?

$m$ is the gradient (steepness and direction — positive slopes up left to right, negative slopes down, zero is horizontal); $c$ is the y-intercept, the point $(0,c)$ where the line crosses the $y$ -axis. Knowing these lets you predict roughly what the line should look like before plotting — a useful check against silly errors.

02 — Explanation

How do I plot a line using a table of values?

Choose at least three $x$ -values across the given range (always include $x=0$ if possible, since it instantly confirms $c$ ).
Substitute each into the equation to get the matching $y$ -values.
Plot the coordinates as small crosses and join them with a ruler, extending fully across the required range.

Example: $y=3x-1$ at $x=-2,0,3$ gives $(-2,-7)$ , $(0,-1)$ , $(3,8)$ .

Exam Tip: Always plot a third point beyond the two you need. If all three fall on a straight line, that confirms your arithmetic was correct — if one is off the line, you know exactly where to recheck.

03 — Explanation

What if the equation isn't already in the form y=mx+c?

Rearrange to make $y$ the subject first.

Example: $3x-2y=-6$ . Subtract $3x$ : $-2y=-3x-6$ . Divide by $-2$ : $y=1.5x+3$ , so $m=1.5$ , $c=3$ .

04 — Explanation

What do horizontal and vertical lines look like?

$y=b$ is a horizontal line through $(0,b)$ — gradient $0$ . $x=a$ is a vertical line through $(a,0)$ — not a function of $x$ at all, just every point with that fixed $x$ -coordinate.

Common Misconception: Confusing $x=a$ (vertical) with $y=a$ (horizontal) — remember $x=a$ fixes the horizontal position, so the line itself must run vertically.

05 — Explanation

How do I use the gradient to plot a line without a table?

Plot the $y$ -intercept $(0,c)$ first, then use the gradient as "steps": move $1$ right, $m$ up (down if negative). For a fractional gradient like $m=\frac23$ , step $3$ right and $2$ up instead, to land on whole-number points.

Exam Tip: Don't judge a gradient by eye — if the axes use different scales, a gradient of 2 can look shallow or steep depending on the scale. Always count actual units, never squares, and trust the calculation over the picture.

06 — Explanation

How do I find the x-intercept (root) of a line?

Set $y=0$ and solve for $x$ .

Example: $y=x-5$ . Set $0=x-5$ , so $x=5$ — the line crosses the $x$ -axis at $(5,0)$ .

07 — Explanation

How do I choose sensible axes and scales if they're not given?

Work out the minimum and maximum $y$ -values across the required $x$ -range first, then choose axis bounds slightly beyond those extremes, using "easy" scale intervals like 1, 2, 5, or 10 per square so the line is quick to plot and read.

Example: For $y=3x+5$ with $-4\le x\le3$ : $y$ ranges from $3(-4)+5=-7$ up to $3(3)+5=14$ . A sensible choice is $x$ from $-5$ to $4$ , and $y$ from $-10$ to $15$ , labelling both axes clearly.

Common Misconception: Using the same scale on both axes regardless of how different the $x$ - and $y$ -ranges are. It's perfectly fine — and often necessary — for 1 square to represent a different number of units on each axis, as long as you label this clearly.

08 — Worked example

Worked Example

Draw the graph of $y=2x+1$ for $-2 \le x \le 3$ , and use it to find the value of $x$ when $y=4$ .

Table of values: $x=-2 \to y=-3$ ; $x=0 \to y=1$ ; $x=3 \to y=7$ .
Plot $(-2,-3)$ , $(0,1)$ , $(3,7)$ and join with a ruler.
Reading across from $y=4$ to the line, then down to the $x$ -axis, gives $x=1.5$ (check algebraically: $4=2x+1 \Rightarrow x=1.5$ ✓).