Algebra - Graphs: plot graphs of equations that correspond to straight-line graphs in the coordinate plane; use the form y = mx + c to identify parallel and perpendicular lines; find the equation of the line through two given points or through one point with a given gradient (Higher tier)

The idea

Higher tier only; the addition to shared content is perpendicular lines. In y = mx + c, m is the gradient and c the y-intercept. Parallel lines share the same gradient; perpendicular lines have gradients multiplying to −1, so each is the negative reciprocal of the other. A vertical line has no gradient and is perpendicular to a horizontal one. Rearrange so y stands alone before comparing: 3y = 6x + 5 has gradient 2, not 6. To plot a straight-line graph, substitute two or three x-values, plot the points and join them with a ruler. Through two points, find the gradient as the change in y over the change in x, then substitute one point to get c. For y = 4x − 1 through (8, 5), the perpendicular gradient is −1/4, and 5 = −2 + c gives c = 7, so y = −x/4 + 7.

What earns the marks

• Plotting a straight-line graph by substituting x-values, plotting the points and joining them with a ruled line.
• Reading the gradient from y = mx + c only after rearranging so that y is on its own.
• Identifying parallel lines by equal gradients, and perpendicular lines by gradients multiplying to −1.
• Using the negative reciprocal as the gradient of a perpendicular line, even if later work is wrong.
• Finding the gradient through two points as the change in y divided by the change in x, then substituting one point to find c.
• The finished equation, written in whichever form the question asks for.

Common mistakes and how to put them right

• Turning the gradient upside down but leaving its sign alone, so pairing gradient 5 with 1/5 instead of −1/5.

• Changing the sign but not turning the gradient upside down, so pairing gradient 5 with −5.

• Reading the gradient straight off an equation such as 2y = 8x + 3 without dividing through first.

• Keeping the intercept of the original line instead of substituting the point the question gives.

• Joining plotted points freehand or with too few points, so the line is not straight.

Exam tips

• Write the perpendicular gradient as a fraction and cancel it, so −4/8 becomes −1/2 before you substitute.
• Check the pair by multiplying the two gradients together, because anything other than −1 means one of them is wrong.
• If a particular form of the equation is asked for, rearrange as the very last step so the method marks are already safe.