Mathematics 0580 · Syllabus tracker
Graphical Solutions
Solving equations from graphs, finding the line to draw, simultaneous equations from graphs and gradients from tangents.
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Quick revision
Graphical Solutions on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Solving equations with graphs, finding the line to draw, simultaneous equations and gradients from tangents.
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.
- Draw y = 4.
- 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
- Write the curve's expression (e.g. x2 − 3x).
- Rearrange the new equation to: that expression = something.
- Draw y = something.
- 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?
- 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?
- 2x + 1 = 7 − x → x = 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
- Draw the tangent with a ruler.
- Choose two points on it, far apart.
- Gradient = change in y ÷ change in x.
Worked example
A tangent passes through (1, 3) and (4, 12). Find its gradient.
- Rise = 12 − 3 = 9.
- Run = 4 − 1 = 3.
Gradient 3
✗Common mistakes
| Don’t write | Write 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.