Anatomy of a parabola
Every parabola has these features:
| Feature | Description |
|---|---|
| Vertex | the turning point: a minimum if it opens up, a maximum if it opens down |
| Axis of symmetry | the vertical line x = h through the vertex |
| Direction of opening | up (a > 0) or down (a < 0) |
| y-intercept | where it crosses the y-axis |
| Zeros (x-intercepts) | 0, 1 or 2 points where y = 0 |
| Domain | all real numbers |
| Range | y ≥ k (opens up) or y ≤ k (opens down) |
Symmetry
The parabola is a mirror image across its axis of symmetry. Points at equal horizontal distances from the axis have the same y-value.
If the zeros are 1 and 7, the axis is x = 4 (the midpoint).
The role of a
| a | Effect |
|---|---|
| a > 0 | opens up |
| a < 0 | opens down |
| |a| > 1 | vertical stretch: narrower |
| 0 < |a| < 1 | vertical compression: wider |
Graphing from vertex form
y = −2(x − 1)² + 8
- Vertex: (1, 8). Opens down. Maximum 8.
- Axis: x = 1.
- From the vertex, step 1 right and 1 left: y drops by a × 1² = 2 → points (0, 6) and (2, 6).
- Step 2 away: y drops by a × 4 = 8 → points (−1, 0) and (3, 0). These are the zeros!
- Draw a smooth U-shape.
Step pattern for y = x²: from the vertex, move 1 over, 1 up; 2 over, 4 up; 3 over, 9 up. Multiply the "up" amounts by a.
Domain and range
y = −2(x − 1)² + 8
- Domain: all real numbers
- Range: y ≤ 8
Real-world parabolas
- The path of a thrown ball (ignoring air resistance)
- Water from a fountain
- Satellite dishes and car headlights (parabolic reflectors focus signals or light)
- Suspension bridge cables (approximately)
Interpreting features
For h(t) = −4.9t² + 14.7t + 2 (height of a ball):
- y-intercept 2: starting height 2 m
- vertex: maximum height and when it happens
- positive zero: when the ball lands
Worked examples
Example 1: y = 3(x + 2)² − 5: vertex, direction, range? (−2, −5), opens up, y ≥ −5.
Example 2: Zeros at −3 and 5. Axis? x = 1.
Example 3: Which is wider, y = 0.5x² or y = 4x²? y = 0.5x².
Common mistakes
- Confusing narrower and wider with the size of a.
- Writing the range as all real numbers.
- Wrong vertex sign from (x + h).
Try it yourself
- Vertex and range of y = −(x − 4)² + 3?
- A parabola has zeros at 2 and 10. Axis of symmetry?
- y = 2(x − 1)² − 8. Find the zeros.
- Is y = −0.25x² wider or narrower than y = −x²?
Answers: 1) (4, 3); y ≤ 3 2) x = 6 3) x = 3 and x = −1 4) wider
Quick recap
- A parabola has a vertex, an axis of symmetry and a direction of opening.
- a controls direction and width (vertical stretch or compression).
- Use symmetry and the step pattern to graph.