Build a quadratic function from its features
Use a vertex, intercepts or another point to determine a quadratic model in a suitable factored or vertex form.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which form shows the vertex directly?
What you will learn
- Find a quadratic from its vertex and another point
- Find a quadratic from intercepts and a point
- Use the simpler form when the leading coefficient is one
Choose a form that uses the information directly
If the vertex is (h, k), write y = a(x − h)² + k. Substitute a known point to find a.
If the x-intercepts are r and s, write y = a(x − r)(x − s). Another point determines the scale a. When a = 1 is specified, the vertex or intercept data can determine the rule directly.
Check every stated feature
Substitute the given vertex or intercepts into the finished expression. Vertex form shows the turning point; factored form shows the roots.
The value of a controls opening and vertical stretch. Preserve signs in coordinate values inside brackets.
Pause and explain
A parabola has roots 1 and 4. Which factor form applies?
Worked example
Find the quadratic with vertex (2, −3) passing through (4, 5).
Show the worked solution
- Use y = a(x − 2)² − 3.
- Substitute (4, 5): 5 = a(2)² − 3, so 8 = 4a.
- Therefore a = 2.
Answer y = 2(x − 2)² − 3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find a quadratic with roots 1 and 4 that passes through (0, 8).
Give me a hint
Start with a(x − 1)(x − 4) and substitute the point.
Compare my reasoning
- 8 = a(−1)(−4) = 4a.
- So a = 2.
- The function is y = 2(x − 1)(x − 4).
y = 2(x − 1)(x − 4).
Look for these in your work
- Intercept factors use correct signs
- Point substituted
- Scale factor solved
Find the quadratic with vertex (−1, 2) passing through (1, −6), then state its opening direction.
Give me a hint
Use vertex form and substitute the second point to find a.
Compare my reasoning
- y = a(x + 1)² + 2.
- −6 = 4a + 2, so a = −2.
- The negative a means it opens downward.
y = −2(x + 1)² + 2; it opens down.
Look for these in your work
- Vertex form correct
- Scale factor calculated
- Opening inferred from sign
A parabolic arch has highest point (0, 8) and meets the ground at x = −4 and x = 4. Find a model and its height at x = 2.
Give me a hint
Use intercept form and the vertex to find the scale.
Compare my reasoning
- Use y = a(x − 4)(x + 4) = a(x² − 16).
- At x = 0, 8 = −16a, so a = −1/2.
- At x = 2, y = −1/2(4 − 16) = 6.
y = −0.5x² + 8; height 6 units at x = 2.
Look for these in your work
- Symmetric intercepts represented
- Vertex used to find a
- Prediction substituted
Common mistakes
- Using the vertex coordinates with wrong signs
- Forgetting the scale factor a
- Assuming the curve opens upward
- Not checking the given point
What can you explain now?
What vertex is shown by y = −(x + 1)² + 4?
Compare with the explanation
(−1, 4)
Compare with a(x − h)² + k.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
When are vertex form and intercept form the quicker choice, and how can a known point determine a?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.