Algebra - Graphs: identify and interpret gradients and intercepts of linear functions graphically and algebraically

The idea

The gradient of a straight line says how steep it is: the change in y for every increase of 1 in x. Read it from a graph by choosing two points that sit on grid intersections and dividing the rise by the run, so a line running from (1, 2) to (5, 14) rises 12 across a run of 4 and has gradient 3. A line that falls from left to right has a negative gradient. The y intercept is the value of y where the line meets the y axis, which is the number on its own once the equation has y by itself. In a context both numbers carry meaning: for a taxi fare of y = 2x + 3, the 3 is a fixed charge in pounds and the 2 is the cost of each extra mile. The x intercept is where the line crosses the x axis, found by setting y = 0.

What earns the marks

• A correct rise and run taken from the graph using two points on grid intersections.
• The gradient as a number, carrying a minus sign when the line falls from left to right.
• The intercept read from the point where the line meets the y axis.
• Interpreting the gradient in the units of the context, such as pounds per mile or metres per second.

Common mistakes and how to put them right

• Counting squares instead of units, so the gradient is out by the scale of the axis.

• Dividing the run by the rise.

• Giving a positive gradient for a line that slopes downwards.

• Reading the intercept from the bottom left corner of the grid when the axes do not meet at the origin.

Exam tips

• Choose two points that are far apart and land on grid intersections, so the arithmetic stays exact.
• Give the gradient its units whenever the question is set in a context, because the interpretation mark needs them.
• If y is not on its own in the equation, rearrange before you name the gradient.