What you will be able to do
- The sign of a tells whether the parabola opens up or down.
- The vertex lies on the axis of symmetry.
- Zeros are x-intercepts when real.
- Pairs of points equally spaced from the axis have equal y-values.
Read the key features
| Feature | Meaning | How to find |
|---|---|---|
| Opening | Upward or downward | Sign of leading coefficient a |
| Axis of symmetry | Vertical line through vertex | x = −b/(2a) in standard form |
| Vertex | Maximum or minimum point | Evaluate the function at the axis |
| x-intercepts | Zeros or roots | Solve f(x) = 0 |
| y-intercept | Value at x = 0 | Evaluate f(0) |
Worked graph from standard form
Worked example 1
Graph y = x² − 4x + 3.
- 1.a = 1, so the graph opens upward.
- 2.Axis: x = −(−4)/(2·1) = 2.
- 3.Vertex: f(2) = 4 − 8 + 3 = −1, so (2, −1).
- 4.Factor: (x − 1)(x − 3), so x-intercepts are (1, 0) and (3, 0).
- 5.y-intercept is (0, 3); by symmetry (4, 3) also lies on the graph.
Answer: Vertex (2, −1), axis x = 2, zeros 1 and 3
Check: The intercepts are symmetric around x = 2.
Graph from vertex form
Worked example 1
Graph y = −2(x + 1)² + 8.
- 1.Vertex form gives vertex (−1, 8).
- 2.The graph opens downward because a = −2.
- 3.Axis is x = −1.
- 4.One unit from the axis, y = 6 at x = 0 and x = −2.
Answer: Vertex (−1, 8), downward opening
Check: Symmetric points (0, 6) and (−2, 6).
Check graph accuracy
Practise quadratic equationsLearn factoring polynomials
- Label a consistent scale.
- Plot the vertex and axis before sketching.
- Use symmetric point pairs.
- Confirm intercepts by substitution.
- Connect points with a smooth curve, not straight segments.
- Respect a contextual domain if one is given.
When independent practice is not enough
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Useful answers
Frequently asked questions
What is the axis of symmetry?
It is the vertical line through the vertex that divides the parabola into mirror-image halves.
Can a quadratic have no x-intercepts?
Yes. A parabola can remain entirely above or below the x-axis, so it has no real zeros.
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Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.