Quadratic Graphs: How to Plot and Interpret Parabolas

A quadratic graph is the curve produced when you plot y = ax² + bx + c. The shape is called a parabola. When a is positive the parabola is U-shaped (opens upward); when a is negative it is ∩-shaped (opens downward). Quadratic graphs appear on both Foundation and Higher GCSE papers.

What does a quadratic graph look like?

Every quadratic graph is a smooth, symmetrical curve — a parabola. Key features to identify:

Roots (x-intercepts): where the parabola crosses the x-axis (y = 0). A quadratic can have 0, 1, or 2 roots.

Vertex (turning point): the lowest point of a U-shaped parabola (minimum) or the highest point of a ∩-shaped parabola (maximum).

Line of symmetry: a vertical line passing through the vertex. For y = ax² + bx + c, the line of symmetry is x = −b ÷ (2a).

y-intercept: where the graph crosses the y-axis. This is always at y = c (set x = 0).

How do you build a table of values for a quadratic graph?

Step-by-step method:

Write out the x-values given in the question (e.g. x = −3 to x = 3).

Substitute each x-value into the equation and calculate y.

Check for errors by confirming the table is roughly symmetrical about the vertex.

Plot the (x, y) pairs on the grid.

Join the points with a smooth curve — not straight lines between them.

Worked example — plot y = x² − 2x − 3 for x = −2 to x = 4:

x | x² | −2x | −3 | y
−2 | 4 | 4 | −3 | 5
−1 | 1 | 2 | −3 | 0
0 | 0 | 0 | −3 | −3
1 | 1 | −2 | −3 | −4
2 | 4 | −4 | −3 | −3
3 | 9 | −6 | −3 | 0
4 | 16 | −8 | −3 | 5

How do you identify the roots from the table or graph?

The roots are the x-values where y = 0. From the table above, y = 0 at x = −1 and at x = 3. These are the two roots: x = −1 and x = 3.

On the graph these appear as the two points where the parabola crosses the x-axis. You can read off solutions to the quadratic equation x² − 2x − 3 = 0 directly from the graph without any algebra.

How do you find the vertex and line of symmetry?

From the table: The vertex occurs at the minimum y-value. In the table above, the minimum is y = −4 at x = 1.

From the formula: Line of symmetry is x = −b ÷ (2a) = −(−2) ÷ (2 × 1) = 2 ÷ 2 = 1.

Substitute x = 1 into the equation: y = 1 − 2 − 3 = −4.

So the vertex (minimum point) is (1, −4) and the line of symmetry is x = 1.

Notice from the table that the y-values are symmetric about x = 1: y at x = −2 equals y at x = 4 (both 5), and y at x = −1 equals y at x = 3 (both 0). This symmetry is a useful check.

How do you use a quadratic graph to solve equations?

A quadratic graph helps you solve equations graphically. For example:

Solve x² − 2x − 3 = 0: read the x-values where the graph crosses y = 0. Answer: x = −1 and x = 3.

Solve x² − 2x − 3 = 2: draw the horizontal line y = 2 on the same axes, then read the x-values where the parabola crosses that line. From the table, y = 5 at x = −2 and x = 4, and y = 0 at x = −1 and x = 3, so y = 2 lies somewhere between; read the x-values from the graph (approximately x ≈ −1.4 and x ≈ 3.4).

What common mistakes should you avoid?

Connecting points with straight lines. A quadratic curve is smooth; straight segments between points do not produce a parabola and will lose marks.

Arithmetic errors in the table. Recalculate at least two y-values to check. A single wrong entry produces an obvious kink in the curve.

Misreading the roots. Roots are x-values, not y-values. They occur where y = 0, i.e. on the x-axis.

Forgetting the y-intercept. Set x = 0 to find the y-intercept quickly as a check: for y = x² − 2x − 3, y-intercept = −3. It should match your table.

Frequently asked questions

What if the parabola does not cross the x-axis?

If the parabola does not touch the x-axis, the quadratic equation y = 0 has no real solutions. This happens when the discriminant b² − 4ac is negative. On the graph, a U-shaped parabola whose vertex is above the x-axis has no roots. A ∩-shaped parabola whose vertex is below the x-axis also has no roots.

How is the coefficient of x² linked to the shape of the parabola?

If the coefficient of x² (the value of a) is positive, the parabola is U-shaped with a minimum turning point. If a is negative, the parabola is ∩-shaped with a maximum turning point. The larger the value of |a|, the narrower (steeper) the parabola; smaller |a| produces a wider curve.

Can I use a quadratic graph to find the equation of the parabola?

Yes — if you can read the roots and one other point (such as the y-intercept), you can write the equation. If roots are at x = p and x = q, the equation is y = a(x − p)(x − q). Substitute a known point to find a.

What is the difference between a root and a solution of a quadratic?

They are the same thing. The roots of y = ax² + bx + c are the x-values where y = 0 — these are also the solutions to the equation ax² + bx + c = 0. The word "root" is used in the context of equations; "x-intercept" or "zero" refers to the graph.

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