A parabola is the U-shaped graph of a quadratic function, usually written as y = ax^2 + bx + c. Graphing parabolas helps you see key features such as the vertex, axis of symmetry, intercepts, direction, and width. These features make it easier to sketch the graph accurately without plotting many points.
Parabolas appear in physics, engineering, architecture, and any situation involving quadratic change, such as projectile motion.
Key Facts
- Standard form of a quadratic function: y = ax^2 + bx + c
- Vertex form: y = a(x - h)^2 + k, where the vertex is (h, k)
- Axis of symmetry: x = -b/(2a) for y = ax^2 + bx + c
- If a > 0, the parabola opens upward; if a < 0, it opens downward
- The y-intercept of y = ax^2 + bx + c is (0, c)
- The x-intercepts are the solutions to ax^2 + bx + c = 0
Vocabulary
- Parabola
- A parabola is the curved U-shaped graph of a quadratic function.
- Vertex
- The vertex is the highest or lowest point of a parabola.
- Axis of Symmetry
- The axis of symmetry is the vertical line that divides a parabola into two matching halves.
- Intercept
- An intercept is a point where a graph crosses the x-axis or y-axis.
- Quadratic Function
- A quadratic function is a function that can be written in the form y = ax^2 + bx + c where a is not zero.
Common Mistakes to Avoid
- Using the wrong sign for the vertex in vertex form is a common mistake. In y = a(x - h)^2 + k, the x-coordinate of the vertex is h, not -h.
- Forgetting that the axis of symmetry is a vertical line leads to incorrect graph labels. It should be written as x = value, not y = value.
- Assuming every parabola has two x-intercepts is wrong. A parabola can have two, one, or zero x-intercepts depending on how it meets the x-axis.
- Changing a without changing the shape is incorrect. A larger absolute value of a makes the parabola narrower, while a smaller absolute value of a makes it wider.
Practice Questions
- 1 For y = x^2 - 4x + 3, find the vertex, axis of symmetry, y-intercept, and x-intercepts.
- 2 Graph y = -2(x + 1)^2 + 8 by identifying the vertex, direction of opening, axis of symmetry, and at least two symmetric points.
- 3 Two parabolas have the same vertex at (0, 0). One is y = x^2 and the other is y = 4x^2. Explain which graph is narrower and why.