Key Concepts
- Quadratic functions take the form (f(x) = ax^2 + bx + c).
- The parabola formed by a quadratic function can have intercepts and a vertex that describe its behavior.
- The x-intercepts of a quadratic function are the solutions to (ax^2 + bx + c = 0).
- The x-coordinate of the vertex of (f(x) = ax^2 + bx + c) is given by (x = -\frac{b}{2a}).
- The two x-intercepts of a quadratic function are symmetric about the vertex.
- If a quadratic function is given by (f(x) = a(x - p)(x - q)), its zeros (x-intercepts) are (p) and )q).
- A quadratic function can be expressed in vertex form (f(x) = a(x - h)^2 + k) where ((h, k)) is the vertex.
Skills Covered
- Finding the x-intercepts and y-intercepts in a quadratic graph.
- Finding the axis of symmetry using (x = -\frac{b}{2a}).
- Using the vertex as a midpoint to find the second x-intercept.
- Factoring quadratic expressions (ax^2 + bx + c) and determining the x-intercepts.
- Completing the square to rewrite functions in vertex form.
Example Problems
- Solve (x^2 - 5x + 6 = 0) by factoring to find the x-intercepts.
- A ball follows the equation (h(t) = -5t^2 + 20t + 3). What is the maximum height?
- Given that a quadratic function has vertex ((-2, 5)) and one x-intercept at 4, determine the other x-intercept.
- Rewrite (x^2 + 6x + 8) in vertex form and find the vertex.