0580

Further Graphs and Tangents

Coordinate Geometry and Graphs

Drawing the tangent

  • A curve has a different gradient at every point, and the gradient at one point is the gradient of the tangent there
  • A tangent is a straight line that touches the curve at that point without cutting across it
  • Rest a ruler against the curve at that point and extend the line right across the grid, so the triangle you read from is large
The curve y = x² − 9x + 8 with a tangent drawn at (1, 0) sloping steeply down and another at (6, −10) sloping up, showing that the gradient of the curve at a point equals the gradient of the tangent there
Source: Finding gradients of tangents by Save My Exams
  • Read two points far apart on the tangent, ideally on grid intersections, and find the vertical change divided by the horizontal change
    • A tangent passing through (1, 2) and (5, 14) has gradient 12 ÷ 4 = 3
A tangent drawn to a falling curve at x = 4, with a right-angled triangle marked on the tangent giving a rise of 2.5 against a run of 4, so the gradient at that point is about −0.6
Source: Finding gradients of tangents by Save My Exams
  • Going downhill gives a negative gradient, so put the sign in yourself
  • The answer is only an estimate, because the tangent is positioned by eye

What the gradient means

  • The gradient is the rate at which the vertical quantity changes as the horizontal one increases by 1
  • Its units are the units on the vertical axis divided by the units on the horizontal axis
  • On a distance–time graph that rate is the speed, and on a speed–time graph it is the acceleration
  • Answering "what does the gradient represent" needs the quantity named, with its units, not just a number
Exam tip

Never rule a curve and never join the plots with straight segments — both lose the drawing marks. Plot to within half a square, leave the x = 0 cell of a reciprocal table empty, and bracket negative values before squaring. When solving from a graph, show the line you decided to draw: that choice is where the mark sits, not the reading.

Worked example

Reading a tangent off the grid

A tangent drawn to a curve at x = 3 passes through the grid points (1, 9) and (6, −1). Estimate the gradient of the curve at x = 3.

Solution:

  • Vertical change from the first point to the second: −1 − 9 = −10
  • Horizontal change over the same span: 6 − 1 = 5
  • Divide the two: −10 ÷ 5 = −2
  • The gradient of the tangent is −2, and the curve has the same gradient where the tangent touches it
  • So the gradient of the curve at x = 3 is approximately −2
  • The tangent slopes downhill, which agrees with the negative answer