A quadratic graph has the shape of a smooth U-shaped curve called a parabola. For KS3 maths you will plot graphs such as y = x², y = x² − 3 and y = 2x², using a table of values. Every quadratic graph has a turning point called the vertex and a vertical line of symmetry through it.
What makes a graph quadratic?
A graph is quadratic when the highest power of x in its equation is x². The general form is y = ax² + bx + c, where a, b and c are numbers and a ≠ 0.
At KS3 you will mainly work with simpler quadratics:
| Equation | Shape |
|---|---|
| y = x² | U-shape with vertex at (0, 0) |
| y = x² + 3 | Same U-shape shifted up 3 units |
| y = x² − 5 | Same U-shape shifted down 5 units |
| y = 2x² | Narrower U-shape (steeper sides) |
| y = −x² | Upside-down U (n-shape) — a is negative |
When a > 0 the parabola opens upwards (U-shape). When a < 0 it opens downwards (n-shape).
How do you complete a table of values for y = x²?
Worked example 1: Plot y = x² for x values from −3 to 3.
Step 1: Set up the table. Square each x value — remember that squaring a negative gives a positive.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| x² | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
| y | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
Step 2: Plot the seven points: (−3, 9), (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4), (3, 9).
Step 3: Join them with a smooth curve — not straight lines. The curve should look like a symmetrical U.
How do you complete a table of values for y = x² − 2x − 3?
Worked example 2: Plot y = x² − 2x − 3 for x from −2 to 4.
Calculate y for each x by substituting:
| 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 |
The lowest point (vertex) is at x = 1, y = −4, giving the coordinate (1, −4). The graph crosses the x-axis at x = −1 and x = 3.
How do you find the vertex and line of symmetry?
The vertex is the turning point of the parabola — the lowest point for a U-shape or the highest for an n-shape. It sits on the line of symmetry, which is a vertical line that mirrors the two sides of the curve.
For a U-shaped quadratic, the vertex is the minimum point. For an n-shaped quadratic, it is the maximum point.
To find the line of symmetry from a table of values: look for the x-value where y is smallest (or largest). In Worked example 2, the minimum is y = −4 at x = 1, so the line of symmetry is x = 1.
You can also find it by noting that the two x-values where the graph crosses the x-axis (the roots) are symmetric. The roots are x = −1 and x = 3; the midpoint is (−1 + 3) ÷ 2 = 1, confirming x = 1.
How do you plot the graph and draw the curve?
Use these steps every time:
- Draw axes. Choose a scale that fits all your y-values comfortably.
- Plot each coordinate from your table as a clear dot.
- Join with a smooth curve — lift your pencil off the paper and draw one continuous, rounded line through all points.
- Do not use a ruler. A parabola is curved, not straight.
- Label the graph with its equation (e.g. y = x² − 2x − 3).
A common error is connecting points with straight line segments. This creates a zigzag that looks like two straight lines — quite different from the smooth parabola expected in the mark scheme.
How do you use a quadratic graph to solve an equation?
Once the graph is drawn, you can use it to estimate solutions.
Example: Use the graph of y = x² − 2x − 3 to solve x² − 2x − 3 = 0.
The solution to any equation y = k is found where the graph reaches the height y = k. For y = 0, that is where the curve crosses the x-axis.
From the table and graph, the crossings are at x = −1 and x = 3. So the solutions are x = −1 and x = 3.
At GCSE you will also use the graph to solve related equations (such as x² − 2x − 3 = 2) by drawing a horizontal line at y = 2 and reading off the x-values where it intersects the curve.
Frequently asked questions
Why does squaring a negative number give a positive result?
Because multiplying a negative number by itself produces a positive: (−3) × (−3) = +9. This is why the left and right sides of a parabola are mirror images — x = 3 and x = −3 give the same y-value. Forgetting this rule leads to incorrect table values, with the left side of the table going negative when it should mirror the right.
What is the difference between a quadratic graph and a cubic graph?
A quadratic graph (y = x²) has exactly one turning point and is U-shaped or n-shaped. A cubic graph (y = x³) has an S-shaped curve and typically two turning points. The key visual difference: quadratics are symmetric about a vertical line; most cubics are not.
Do I always need to use x values from −3 to 3?
Not necessarily — the question will usually specify the range of x values to use. Always use the x values given in the question. If no range is specified, a range of about 5 or 7 x-values centred on the vertex gives a clear picture.
Can I check my graph is correct?
Yes. The curve must be symmetric about the line of symmetry, so if (−2, 4) is on your graph, (2, 4) should also be on it (for y = x²). If the two sides do not mirror each other, recheck your table of values for arithmetic errors.
For guided KS3 algebra and graph practice with Professor Pi — visit aitutors.me.