Functions and graphs

For G3 Math graphs, start with the axes and scales, then identify the gradient, intercepts and any turning point. A straight line has constant gradient; a curve usually does not. The examples below connect equations to the features you need to read or sketch.

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.

Which graph feature do you need?

Straight line y = mx + c: m is the gradient; (0, c) is the y-intercept. Use two points to check rise divided by run.
Quadratic y = a(x − p)² + q: The turning point is (p, q). The sign of a determines whether it is a minimum or maximum.
Intersection: The same coordinate satisfies both equations. This connects graphs to simultaneous equations.
Curve gradient: Draw a tangent at the requested point and calculate the gradient of that tangent.

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.

Common 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.

Related G3 Math guides

You may need first

These verified prerequisite links suggest useful foundations. They do not make a claim about your current readiness.

This unlocks

This guide's skills appear in these verified connections to other published guides.

Simultaneous equationsFor the graphical method, the common solution is the intersection of two lines. Read both coordinates.

Choose a useful next resource

Choose one step that matches the gap you just found.

Interpret common graph-reading errors Explore foundations connected to functions and graphs