Straight line graphs

Hold c still, change m

This line is y = mx + 2, so it always crosses the y-axis at (0, 2). Lift the dot to change m and watch the whole line pivot about that fixed point.

Maths · Straight-line graphs

Which way does it lean?

Put each line in the category that describes its gradient.

8 of 8 still to sort.

Some lines are given as equations, some as two points. Decide where each belongs, then read why.

Predict, then check

Do not rearrange anything yet. Trust your first instinct, then we will test it.

The line is y = 7 − 4x. What are its gradient and y-intercept?

What you need to know

• A relationship is linear if its graph is a straight line.
• Draw one by substituting x values and plotting the (x, y) pairs. The y values go up by a common difference.
• A straight line has infinitely many points, all fitting its equation, so draw it right to the edge of the graph.
• Two points are enough to draw it, but checking a third point catches errors.
• For ax + by = c, substitute x = 0 and y = 0 to get two easy points.
• Have a goTry it on 2x + 5y = 10. Which two points do you get from x = 0 and y = 0?

  (0, 2) and (5, 0)

  Putting x = 0 leaves 5y = 10, so y = 2. Putting y = 0 leaves 2x = 10, so x = 5. Each substitution removes one term.

• In y = mx + c, m is the gradient and (0, c) is the y-intercept, where the line crosses the y-axis.
• Gradient is the change in y for each +1 in x: positive if y rises, negative if it falls, zero if flat.
• Gradient from two points = change in y ÷ change in x. Same x value? That divides by 0, so no gradient.
• Have a goWhat is the gradient of the line through (1, 9) and (3, 3)? Watch the sign.

  −3

  The change in y is 3 − 9 = −6 and the change in x is 3 − 1 = 2, so −6 ÷ 2 = −3. y falls as x rises, so it must be negative.

• Rearrange to y = mx + c before reading m and c. The gradient is the coefficient of x, not the first number.
• Have a goSam is very confident that y = 10 − 3x has gradient 10, "because 10 comes first". What do you tell Sam?

  Not so fast, Sam: it is y = −3x + 10, so the gradient is −3.

  The gradient is the coefficient of x once the equation is in the form y = mx + c, and it keeps its sign. The 10 is the y-intercept.

• To find an equation: get m, substitute a point to find c, then check with another point.

The big picture

A straight-line graph is the picture of a linear equation. In y = mx + c, m is the gradient (the change in y for each +1 in x) and c fixes where the line crosses the y-axis, at (0, c). You can draw a line from its equation, read m and c once the equation is rearranged, and build the equation of a line from a gradient and a point or from two points.

⚠ Watch out

Reading the first number as the gradient, or dropping a minus sign. In y = 6 − 5x, the gradient is not 6. Rearrange to y = −5x + 6 and the gradient is −5.

Memory hook

m is how much the line Moves up for each step right. c is where the line Crosses the y-axis.

Check yourself

A line has gradient 4 and passes through (1, 7). Find its equation in the form y = mx + c, then show that (2, 11) is on it.