Functions and graphs
A graph shows pairs of input and output values. Read the axes and scales first. A line has constant gradient; a curve generally has a changing gradient. For a quadratic, the turning point, symmetry and intercepts help you sketch and interpret it.
What this guide covers
Coordinates, linear and quadratic graphs, specified power and exponential graphs, and estimating a curve gradient with a tangent. The checks focus on interpreting coordinates and quadratic features.
Syllabus reference: 2026 Mathematics 4052, N6, page 7. These are Mentora search groupings, not new official syllabus categories.
Worked example
Describe y = (x − 2)² − 9.
The squared term is smallest at x = 2, giving the minimum point (2, −9) and symmetry line x = 2. Setting y = 0 gives (x − 2)² = 9, so the x-intercepts are (−1, 0) and (5, 0). At x = 0, y = −5, so the y-intercept is (0, −5).
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
For y = 3x − 4, find y when x = −2.
Show worked answer
Substitute with the sign: y = 3(−2) − 4 = −10. The point is (−2, −10).
Check 2
Find the turning point of y = −(x + 1)² + 6.
Show worked answer
The square is zero at x = −1. The negative coefficient makes this a maximum, at (−1, 6).
Check 3
A tangent to a curve passes through (1, 2) and (5, 10). Estimate the curve gradient at the tangent point.
Show worked answer
Use two points on the tangent: (10 − 2)/(5 − 1) = 8/4 = 2. This estimates the curve gradient only at the point of tangency.
Mistakes to watch for
- An intercept is a coordinate pair; setting x = 0 finds the y-intercept.
- The turning point of (x − p)² + q is (p, q), not (−p, q).
- For a curve gradient use the tangent, not two widely separated points on the curve.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
Useful foundations to check
These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.
Applications and next topics
Graphs connect equations to intersections and represent rates in context. Coordinate geometry develops straight-line equations and distances from the same coordinate system.
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Interpret common graph-reading errors.
Sources and review
Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-11; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.