Sheet № 147 · Foundation + Higher · AQA · Edexcel · OCR
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
§Key definitions
Question:
Sketch y = x² − 4x + 3.
Answer:
U-shaped parabola with roots at (1, 0) and (3, 0), y-intercept (0, 3), and minimum at (2, −1).
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.
§Formulas to memorise
Roots — set y = 0 and solve ax² + bx + c = 0
y-intercept — set x = 0, giving y = c
Line of symmetry: x = −b/(2a)
Turning point x-coordinate = −b/(2a), then substitute to find y
Determine the shape. — If a > 0 (positive x²), the parabola is U-shaped. If a < 0 (negative x²), it is ∩-shaped.
Find the y-intercept — by substituting x = 0. This gives the point (0, c).
Find the roots — by solving ax² + bx + c = 0 (factorise, use the formula, or complete the square).
Find the turning point. — The x-coordinate is x = −b/(2a). Substitute this into the equation to find the y-coordinate.
Plot these key points — and draw a smooth parabola through them.
Label all key coordinates — on your sketch.
Worked example
Sketch y = x² − 4x + 3.
Working:
⚠ 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.