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Mathematics 4024

Quick revision notes · Mathematics 4024 · Chapter 16 of 35

Graphical Solutions

Solving equations with graphs, finding the line to draw, simultaneous equations and gradients from tangents.

4 sectionsWorked examples Common mistakes
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At a glance

Solving equations with graphs, finding the line to draw, simultaneous equations and gradients from tangents.

  1. Solving equations from a graph
  2. Finding the line to draw
  3. Simultaneous equations on a graph
  4. Gradient of a curve

1Solving equations from a graph

  • Solutions of f(x) = 0 are the x values where the curve crosses the x-axis.
  • Solutions of f(x) = k: draw the horizontal line y = k and read the x values where it meets the curve.
  • The number of crossing points is the number of solutions.

Worked example

The graph of y = x2 − x − 6 is drawn. Solve x2 − x − 6 = 4.

  1. Draw y = 4.
  2. Read the x values where the line meets the curve.

about x = −2.7 and x = 3.7

2Finding the line to draw

  • You have the graph of one curve and must solve a different equation.
  • Rearrange the new equation so one side is exactly the curve you have; the other side is the line to draw.

How to do it

  1. Write the curve's expression (e.g. x2 − 3x).
  2. Rearrange the new equation to: that expression = something.
  3. Draw y = something.
  4. Read the x values of the crossing points.

Worked example

The graph of y = x2 − 3x is drawn. What line should you draw to solve x2 − 5x + 2 = 0?

  1. Add 2x − 2 to both sides: x2 − 3x = 2x − 2.

Draw y = 2x − 2

Exam tip: Check by subtracting: x2 − 3x − (2x − 2) should give back x2 − 5x + 2.

3Simultaneous equations on a graph

  • The solution is the point where the two graphs cross: read both x and y.
  • A line and a curve can cross 0, 1 or 2 times (a cubic up to 3).

Worked example

Where do y = 2x + 1 and y = 7 − x meet?

  1. 2x + 1 = 7 − x → x = 2.
  2. y = 2(2) + 1 = 5.

(2, 5)

4Gradient of a curve

  • Draw a tangent at the point: a straight line that touches the curve there without cutting it.
  • Pick two points far apart on the tangent; gradient = rise ÷ run.
  • On a distance–time graph the gradient is speed; on a speed–time graph it is acceleration.

How to do it

  1. Draw the tangent with a ruler.
  2. Choose two points on it, far apart.
  3. Gradient = change in y ÷ change in x.

Worked example

A tangent passes through (1, 3) and (4, 12). Find its gradient.

  1. Rise = 12 − 3 = 9.
  2. Run = 4 − 1 = 3.

Gradient 3

✗Common mistakes

Don’t writeWrite instead
Reading y values when x values are asked Solutions of an equation in x are x values
Drawing y = x − 3 for x2 + x − 3 = 0 with y = x2 + 2x Add the same to both sides: y = x + 3
Using two points on the curve for a gradient Use two points on the tangent
A tangent that cuts through the curve It must touch at one point only

★In the exam

  • Draw lines long enough to meet the curve everywhere it can.
  • State the equation of any line you draw.
  • Gradient estimates are given a range of accepted answers — draw carefully and read accurately.