Graphs of functions
Recognising and sketching quadratic, cubic, reciprocal and exponential graphs.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.7.1Recognise, sketch and interpret graphs of linear, quadratic, cubic, reciprocal and exponential functions.
6 minute read
Graphs of functions
Recognising a graph from its equation — and vice versa — is quick marks once the shapes are known.
The standard shapes
- Linear
y = mx + c— a straight line. - Quadratic
y = ax² + bx + c— a parabola. Opens upward if a > 0, downward if a < 0. Symmetrical about a vertical line through the turning point. - Cubic
y = ax³ + …— an S-shape, rising overall if a > 0, falling if a < 0. Can have two turning points. - Reciprocal
y = k/x— a hyperbola in two separate parts, with asymptotes at both axes. Never touches either axis, because x cannot be 0 and y cannot be 0. - Exponential
y = aˣ— rises increasingly steeply for a > 1, always passes through (0, 1), and never reaches the x-axis.
Reading key features
- Roots — where the curve crosses the x-axis, found by setting y = 0.
- y-intercept — found by setting x = 0.
- Turning point — the maximum or minimum of a quadratic; it lies on the line of symmetry, midway between the roots.
Solving equations graphically
The x-coordinates where two graphs intersect are the solutions of the equation formed by setting them equal. To solve x² − 3 = x, plot y = x² − 3 and y = x and read off the intersections — or plot y = x² − x − 3 and read the roots. This is why questions often say "use your graph to solve…" — you are expected to read, not to re-derive algebraically.
Think of it like this
Learning graph shapes is like recognising handwriting: once you know the characteristic curve of each family, you can identify the equation at a glance instead of plotting point by point.
Worked examples
Method, step by step
The graph of y = x² − 4x + 3 crosses the x-axis at two points. Find them, and state the equation of the line of symmetry.
- 1Set y = 0: x² − 4x + 3 = 0.
- 2Factorise: (x − 1)(x − 3) = 0, so x = 1 and x = 3.
- 3The line of symmetry lies midway between the roots: x = (1 + 3)/2 = 2.
Roots at (1, 0) and (3, 0); line of symmetry x = 2
Common misconceptions
- Thinking a reciprocal graph eventually touches the axes. It approaches them forever without reaching — that is what an asymptote means.
- Believing every quadratic crosses the x-axis. If the discriminant is negative it has no real roots and floats entirely above or below.
- Assuming an exponential graph passes through the origin. `y = aˣ` passes through (0, 1), since anything to the power 0 is 1.
In the exam
- When sketching, mark the intercepts and the turning point — those are the marked features, not the smoothness of your curve.
- For "use your graph to solve" questions, draw the required line on the graph and read intersections; algebraic re-derivation often earns no marks.