A complete guide to plotting and interpreting quadratic and cubic graphs, finding turning points and roots, solving equations graphically, and shading inequality regions.
Straight-line graphs and function notation are covered in the Algebra article on this site — this guide focuses on what comes next: curved graphs. Quadratic and cubic functions produce distinctive curved shapes with their own key features, and graphs can also be used to solve equations visually or to represent an inequality as a shaded region. This article builds plotting, sketching and interpreting these curved graphs from first principles.
Straight-line graphs (gradient, y = mx + c, equations of lines) and function notation are already covered in depth in the Algebra: Basic, Intermediate & Advanced article on this site. This article picks up from there, focusing specifically on quadratic and cubic curves, and on using graphs to represent inequalities.
To plot y = x² − 2x − 3, first build a table by substituting a range of x-values, then plot the resulting points and join them with a smooth curve (never straight line segments).
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
Every quadratic graph is a parabola — a symmetric curve. Its key features are: the turning point (the minimum or maximum point), the axis of symmetry (a vertical line through the turning point), the y-intercept (where x = 0), and the roots (where the curve crosses the x-axis, i.e. where y = 0).
Completing the square (covered in the Algebra article) directly reveals the turning point: writing y = (x − p)² + q gives a turning point at (p, q).
y = x² − 6x + 5. Completing the square: y = (x − 3)² − 4. The turning point is (3, −4), and since the coefficient of x² is positive, this is a minimum point.
If a quadratic is given in factorised form, y = (x − a)(x − b), its roots are immediately visible at x = a and x = b, and the axis of symmetry lies exactly halfway between them.
y = (x − 2)(x − 8). The roots are at x = 2 and x = 8. The axis of symmetry is at x = (2+8)/2 = x = 5.
A cubic graph (y = ax³ + ...) has a distinctive "S-shaped" curve. If the coefficient of x³ is positive, the curve rises from bottom-left to top-right, generally passing through up to two turning points (a local maximum and a local minimum) along the way.
The plotting method for a cubic is identical to a quadratic: build a table of values across a suitable range, plot the points, and join them with a single smooth curve.
For y = x³ − 3x, some values are: x = −2 gives y = −2; x = −1 gives y = 2; x = 0 gives y = 0; x = 1 gives y = −2; x = 2 gives y = 2. Plotting these and joining smoothly produces the characteristic S-shape.
The solutions (roots) of an equation f(x) = 0 are exactly the x-values where the graph of y = f(x) crosses the x-axis. Reading these crossing points directly off a drawn graph is a valid method for solving an equation, particularly when it cannot be factorised easily.
A graph of y = x² − 2x − 3 crosses the x-axis at x = −1 and x = 3. This tells us the solutions to x² − 2x − 3 = 0 are x = −1 and x = 3.
An inequality such as y > 2x + 1 can be represented on a graph as a shaded region. First draw the boundary line y = 2x + 1 (a solid line for ≤ or ≥, a dashed line for < or >, since the boundary itself is not included), then shade the side of the line that satisfies the inequality — usually checked by testing a convenient point such as (0, 0).
For y > x + 1, test the point (0, 0): is 0 > 0 + 1? No, 0 is not greater than 1. So (0, 0) does not satisfy the inequality, meaning the region to shade is the side of the line not containing (0, 0).
When several inequalities are given together, each boundary line is drawn and shaded (or the unwanted region is shaded instead, depending on convention), and the final answer region is the area satisfying all the inequalities simultaneously — usually left unshaded, or shaded distinctly, if the convention is to shade out the unwanted regions.
Common errors include: joining plotted points with straight line segments instead of a single smooth curve; using a solid line for a strict inequality (< or >) instead of a dashed one; shading the wrong side of a boundary line by skipping the test-point check; and forgetting that the turning point from completed-square form is (p, q) when y = (x − p)² + q, not (−p, q).
| Graph feature | How to find it |
|---|---|
| Quadratic turning point | Complete the square: y = (x−p)² + q gives turning point (p, q) |
| Quadratic roots (factorised form) | y = (x−a)(x−b) has roots at x = a and x = b |
| Solving f(x) = 0 graphically | Read the x-axis crossing points |
| Inequality boundary line | Solid for ≤/≥, dashed for </> |
| Which side to shade | Test a convenient point (often the origin) |
This article is original EDUSAMBAM educational writing. It is designed to extend the straight-line graph and function work already covered in the Algebra article, focusing on quadratic and cubic curves and graphical inequalities. Exact examination requirements can vary between examination boards and syllabuses, so students should also compare their work with the specification and past-paper requirements of their own board.
Recommended study approach: practise plotting quadratic and cubic graphs from a table of values with a smooth curve, then work through identifying turning points and roots before attempting graphical inequality shading with the test-point method.
20 questions covering quadratic and cubic graphs, turning points, roots, solving graphically, and graphical inequalities. Answer every question, then submit to see your score instantly.