Exam Questions - Parabola
Exam Questions - Parabola
The Parabola: Basics and Definitions
- A parabola is a u-shaped curve that consists of points equidistant from a point (focus) and a line (directrix).
- It is defined by the equation y=ax^2+bx+c or x=ay^2+by+c depending on the orientation.
- The vertex of the parabola is the point at which the parabola turns, and it is also the axis of symmetry.
Properties of Parabolas
- Parabolas have a property called focus. It is a point from which distances to the curve are equivalent.
- There is also a directrix, which is a line from which distances to the curve are always equidistant.
- A parabola always opens away from the directrix and towards the focus.
- For a parabola with its axis parallel to the y-axis, the axis of symmetry is the vertical line x = h, where h is the x-coordinate of the vertex.
Solving Parabolic Equations
- To solve parabolic equations, one often needs to make the equation into its standard form, which is y=a(x-h)^2+k, where (h,k) is the vertex of the parabola.
- You can complete the square to put the given equation into the standard form.
- Once in standard form, you can easily find the vertex and axis of symmetry.
- To find the roots (x-intercepts) of a parabola, set y to zero and solve for x. This usually requires factoring, using the quadratic formula, or even completing the square.
Parabolas and Graphing
- When graphing parabolas, the key points to find are the vertex, the focus, and the axis of symmetry.
- If the coefficient of the x^2 term is positive, the parabola opens upwards; if it is negative, it opens downwards.
- The y-intercept is given by the constant term of the quadratic equation.
- The x-intercepts, or points where the parabola crosses the x-axis, are found by setting y=0 in the equation and solving for x.
Example Problem
- Given the equation y=2x^2+3x-2, we want to find the vertex, axis of symmetry and roots of the parabola.
- By completing the square, we get y=2(x+3/4)^2-23/8, which tells the vertex is at (-3/4, -23/8).
- The axis of symmetry is then x=-3/4.
- To find the roots, set y to 0 and solve the equation to get x = 1/2 and -2.
Key notes
- Understanding parabolas and their properties is crucial for a variety of topics in Further Pure 1, including the study of quadratic equations and graphing.
- Practice identifying key characteristics of a parabola from an equation, like the vertex, the focus, and the directrix.
- Review techniques for graphing parabolas and solving parabolic equations, including factoring, completing the square, and the quadratic formula.