Quadratic function

Definition

A quadratic function is a polynomial function of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$ where 'a', 'b', and 'c' are constants, and 'a' is not zero. This type of function produces a parabolic graph that can open either upwards or downwards depending on the sign of 'a'. Quadratic functions have important properties such as vertex, axis of symmetry, and roots, which help in understanding their behavior and applications in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. A quadratic function has at most two real roots, which can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$.
2. The graph of a quadratic function is symmetric about its axis of symmetry, which can be found using the formula $$x = -\frac{b}{2a}$$.
3. The vertex form of a quadratic function is given by $$f(x) = a(x-h)^2 + k$$, where (h, k) is the vertex of the parabola.
4. Quadratic functions can model various real-world situations, such as projectile motion, area optimization, and economics.
5. Completing the square is a method used to convert a quadratic function into vertex form, helping to easily identify its key features.

Review Questions

• How do you determine the vertex of a quadratic function from its standard form?
  • To find the vertex of a quadratic function expressed in standard form $$f(x) = ax^2 + bx + c$$, you can use the formula $$x = -\frac{b}{2a}$$ to find the x-coordinate. Once you have this x-value, plug it back into the function to find the corresponding y-coordinate. This gives you the vertex as the point (h, k), where h is obtained from the x-coordinate and k is found by evaluating f(h).
• Discuss how the discriminant affects the solutions to a quadratic equation.
  • The discriminant, given by $$D = b^2 - 4ac$$ in the context of a quadratic equation $$ax^2 + bx + c = 0$$, determines the nature of the roots. If D > 0, there are two distinct real roots; if D = 0, there is exactly one real root (the parabola touches the x-axis); and if D < 0, there are no real roots (the parabola does not intersect the x-axis). This is essential for understanding how many solutions exist for any given quadratic equation.
• Evaluate how completing the square transforms a quadratic function and its importance in graphing.
  • Completing the square transforms a quadratic function from standard form to vertex form, which is crucial for graphing. The process involves rewriting the function as $$f(x) = a(x-h)^2 + k$$ where (h, k) represents the vertex. This transformation highlights key features such as the vertex position and direction of opening (upward or downward). Understanding this transformation allows for easier graphing and interpretation of how changes in 'a', 'b', and 'c' affect the overall shape and position of the parabola.

Related terms

Vertex: The highest or lowest point of a parabola, depending on whether it opens upwards or downwards, and is crucial for graphing quadratic functions.

Discriminant: A part of the quadratic formula that determines the nature and number of roots of a quadratic equation; calculated as $$D = b^2 - 4ac$$.

Parabola: The U-shaped graph produced by a quadratic function, characterized by its vertex and axis of symmetry.