Sketching Quadratic Graphs

Sketching a quadratic graph means drawing a rough parabola that shows the key features: the roots (where the curve crosses the x-axis), the y-intercept, the turning point, and whether the curve opens upwards or downwards. You do not need to plot every point — just identify the important features and draw a smooth curve through them. This skill appears on both Foundation and Higher papers.

What Is Sketching a Quadratic Graph?

A quadratic graph is the curve you get when you plot y = ax² + bx + c. It is always a parabola. When "sketch" appears in an exam question, it means draw a clearly labelled curve showing key coordinates — it does not need to be perfectly to scale.

Key Formulas

Step-by-Step Method

1. Determine the shape. If a > 0 (positive x²), the parabola is U-shaped. If a < 0 (negative x²), it is ∩-shaped.
2. Find the y-intercept by substituting x = 0. This gives the point (0, c).
3. Find the roots by solving ax² + bx + c = 0 (factorise, use the formula, or complete the square).
4. Find the turning point. The x-coordinate is x = −b/(2a). Substitute this into the equation to find the y-coordinate.
5. Plot these key points and draw a smooth parabola through them.
6. Label all key coordinates on your sketch.

Worked Example 1 — Foundation Level

Question: Sketch y = x² − 4x + 3.

Working:

• Shape: a = 1 > 0, so U-shaped.
• y-intercept: x = 0 gives y = 3. Point: (0, 3).
• Roots: x² − 4x + 3 = 0. Factorise: (x − 1)(x − 3) = 0. Roots at x = 1 and x = 3.
• Turning point: x = −(−4)/(2 × 1) = 2. y = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. Turning point: (2, −1).
• Sketch a U-shaped parabola through (0, 3), (1, 0), (2, −1), (3, 0).

Answer: U-shaped parabola with roots at (1, 0) and (3, 0), y-intercept (0, 3), and minimum at (2, −1).

Worked Example 2 — Higher Level

Question: Sketch y = −x² + 6x − 5.

Working:

• Shape: a = −1 < 0, so ∩-shaped.
• y-intercept: y = −5. Point: (0, −5).
• Roots: −x² + 6x − 5 = 0. Multiply by −1: x² − 6x + 5 = 0. Factorise: (x − 1)(x − 5) = 0. Roots at x = 1 and x = 5.
• Turning point: x = −6/(2 × (−1)) = 3. y = −(9) + 18 − 5 = 4. Maximum at (3, 4).
• Sketch an ∩-shaped parabola through (0, −5), (1, 0), (3, 4), (5, 0).

Answer: ∩-shaped parabola with roots at (1, 0) and (5, 0), y-intercept (0, −5), and maximum at (3, 4).

Worked Example 3 — Exam Style

Question: The quadratic y = x² + 2x − 8 is sketched. Find the coordinates where the graph intersects the axes and the turning point.

Working:

• y-intercept: x = 0 gives y = −8. Point: (0, −8).
• Roots: x² + 2x − 8 = 0. Factorise: (x + 4)(x − 2) = 0. Roots at x = −4 and x = 2.
• Turning point: x = −2/(2 × 1) = −1. y = (−1)² + 2(−1) − 8 = 1 − 2 − 8 = −9. Point: (−1, −9).

Answer: x-intercepts: (−4, 0) and (2, 0). y-intercept: (0, −8). Turning point: (−1, −9).

Common Mistakes

• Drawing a V-shape instead of a curve. A quadratic graph is always a smooth curve (parabola), never a straight-line V.
• Forgetting the y-intercept. Even if the question focuses on roots, you should mark the y-intercept on your sketch.
• Getting the shape wrong. If the coefficient of x² is negative, the parabola opens downwards. Students sometimes draw all parabolas opening upwards.

Exam Tips

• A "sketch" does not require a table of values — just mark and label the key features.
• If the quadratic does not factorise, use the quadratic formula to find the roots or state that the parabola does not cross the x-axis (discriminant < 0).
• The turning point always lies on the line of symmetry, which is exactly halfway between the two roots.
• Mark coordinates clearly. Write (1, 0) rather than just "1" on the x-axis.

Practice Questions

Q1 (Foundation): Find the roots and y-intercept of y = x² − 6x + 8.

Q2 (Foundation): State whether y = −2x² + x + 3 opens upwards or downwards.

Q3 (Higher): Sketch y = x² − 2x − 3 and label all key points.

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Related Topics

• Quadratic Graphs
• Solving Quadratic Equations by Factorising
• Completing the Square

Summary

• A quadratic graph is a smooth parabola (U-shaped if a > 0, ∩-shaped if a < 0).
• Find the y-intercept by setting x = 0, and the roots by setting y = 0.
• The turning point is at x = −b/(2a); substitute back to find the y-coordinate.
• Label all key coordinates on your sketch: roots, y-intercept, and turning point.
• A sketch shows shape and key features — it does not need to be drawn to scale.