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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