Graphs of Functions and Graphical Solutions
Graph questions carry a lot of marks and are usually structured so you can earn most of them: draw the curve, then read things off it. But they are also all-or-nothing in one respect — as tutors put it, one mistake in the intercepts and the whole graph is wrong, taking every later part with it.
1. Recognising the shapes
| Function | Form | Shape |
|---|---|---|
| Linear | y = mx + c | straight line |
| Quadratic | y = ax² + bx + c | parabola — U (a > 0) or ∩ (a < 0) |
| Cubic | y = ax³ + … | S-shaped, up to two turning points |
| Reciprocal | y = k/x | hyperbola — two separate branches |
| Exponential | y = kaˣ | rising (or falling) ever more steeply |
Identify the shape from the highest power before you plot anything. If your plotted points don’t produce that shape, you have made an arithmetic error — this check catches most table mistakes.
A positive x² gives a U; a negative x² gives an ∩. The same logic applies to cubics.
2. Tables of values
Substitute each x into the equation to get y.
For y = x² − 2x − 3:
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
Substitute for x and calculate y — you can’t choose y-values freely. Picking convenient numbers instead of computing them was a documented error.
Negative values need brackets. For x = −2: (−2)² − 2(−2) − 3 = 4 + 4 − 3 = 5. Writing −2² gives −4 on most calculators and wrecks the point.
A built-in check: a quadratic table is symmetrical about its turning point. Here the values run 5, 0, −3, −4, −3, 0, 5 — if yours isn’t symmetric, one entry is wrong.
3. Drawing the curve
Plot with a sharp pencil, and join a curve FREEHAND — not with straight line segments.
- Use graph paper and the scale given
- Plot every point carefully; one misplaced point distorts the shape
- Draw a smooth curve through all points
- A quadratic turning point is rounded, never a sharp corner
Use pencil. Tutors made this point repeatedly — you will want to correct something, and neat graphs score better.
Check the axis scales before plotting. They are often different on each axis.
4. Reading solutions from a graph
This is what the graph is for, and it is where the method marks are.
Solving f(x) = 0 — read the x-intercepts (where the curve crosses the x-axis). These are the roots.
Solving f(x) = 3 — draw the horizontal line y = 3 and read the x-values where it crosses the curve.
Solving two equations simultaneously — read the coordinates of the intersection.
The general method for “use your graph to solve …”:
- Rearrange the equation you’re asked to solve so that one side matches the graph you already drew
- Whatever is left on the other side is the line you must draw
- Read the x-values at the intersections
Example: you have drawn y = x² − 2x − 3 and are asked to solve x² − 2x − 5 = 0.
- Rearrange: x² − 2x − 3 = 2
- So draw the line y = 2 and read off where it meets your curve
This “compare with the graph you have” technique is the single most examined idea here, and it was where students most often got stuck. You are not redrawing anything — you are finding which straight line to add.
If the question says “use your graph”, you must read the answers off it. Solving algebraically instead can score zero, even if the answer is right. Give values to the accuracy the graph allows — usually 1 decimal place.
5. Reciprocal graphs and asymptotes
For y = k/x:
- Two separate branches, in opposite quadrants
- It never touches either axis
An asymptote is a line the curve approaches but never reaches. For y = k/x these are x = 0 (vertical, the y-axis) and y = 0 (horizontal, the x-axis).
x = 0 is vertical and y = 0 is horizontal — mixing these up was a recorded error. The curve has no value at x = 0 because you cannot divide by zero.
Never join the two branches. They are separate; drawing a line across the gap is wrong.
Exponential graphs (y = kaˣ) have a horizontal asymptote at y = 0 and always pass through (0, k), since a⁰ = 1.
6. Gradient of a curve — drawing a tangent
The gradient of a curve changes at every point, so to find it at a particular point you draw a tangent.
Method:
- Mark the point on the curve
- Draw a straight line that just touches the curve there, using a ruler
- Pick two points far apart on that tangent
- Gradient = rise ÷ run
Use a ruler, and take your two points as far apart as possible. A short tangent gives a badly inaccurate gradient. Tutors flagged both points specifically.
Read the coordinates from the tangent line, not from the curve. Beyond the point of contact the tangent and curve separate — using curve points defeats the whole method.
The gradient of a distance–time graph is speed. On a curved distance–time graph, a tangent gives the instantaneous speed at that moment.
7. Key features to identify
| Feature | How to find it |
|---|---|
| y-intercept | where x = 0 — the constant term |
| x-intercepts (roots) | where y = 0 |
| Turning point | the maximum or minimum of the curve |
| Line of symmetry | vertical line through the turning point |
Read the y-intercept where x = 0, not at the edge of the grid. Misreading it as 1.5 instead of −3 was a real error, and everything after it was wrong.
For a maximum or minimum value, read the y-coordinate. For “the value of x that gives it”, read the x-coordinate. The question will make clear which.
8. Mistakes that cost marks
Choosing y-values instead of calculating them.
Squaring negatives without brackets.
Plotting inaccurately, or ignoring the scale.
Joining points with straight lines instead of a smooth curve.
Drawing a sharp point at a turning point.
Solving algebraically when told to use the graph.
Drawing the wrong line when rearranging to solve a new equation.
Joining the branches of a reciprocal graph.
Confusing vertical and horizontal asymptotes.
Drawing a tangent freehand or too short.
Reading tangent coordinates off the curve.
Misreading the y-intercept.
Frequently asked questions
How do I recognise the type of graph? By the highest power: x gives a line, x² a parabola, x³ an S-shape, k/x a hyperbola, aˣ an exponential.
How do I complete a table of values? Substitute each x into the equation, using brackets around negatives.
How do I solve an equation using a graph? Read the x-values where the curve meets y = 0, or where it meets the line you were asked to draw.
How do I use my graph to solve a different equation? Rearrange it so one side matches your graph; the other side is the straight line to draw.
What is an asymptote? A line the curve gets ever closer to but never touches — for y = k/x, the two axes.
Why does a reciprocal graph have two parts? Because it is undefined at x = 0 — you cannot divide by zero.
How do I find the gradient of a curve? Draw a tangent at that point with a ruler, take two widely spaced points on it, and calculate rise ÷ run.
Where is the y-intercept? Where x = 0 — the constant in the equation.
How accurate do my readings need to be? Usually 1 decimal place, to the accuracy the graph allows.
Quick revision checklist
- I can recognise linear, quadratic, cubic, reciprocal and exponential graphs
- I know which way up a parabola opens from the sign of x²
- I can complete a table of values, bracketing negatives
- I use the symmetry of a quadratic table as a check
- I plot accurately in pencil and join with a smooth curve
- I check the axis scales first
- I can read roots and solutions off a graph
- I can rearrange an equation to find which line to draw
- I read answers off the graph when told to
- I know what an asymptote is and can name both for y = k/x
- I never join the branches of a reciprocal graph
- I can draw a tangent with a ruler and find a curve’s gradient
- I take tangent readings from the tangent, not the curve
- I can identify intercepts, turning points and the line of symmetry
These notes cover graphs of functions and graphical solutions in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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