Quadratic function

Definition

A quadratic function is a polynomial function of degree two, generally expressed in the standard form as $$f(x) = ax^2 + bx + c$$ where $$a$$, $$b$$, and $$c$$ are constants and $$a \neq 0$$. This type of function is characterized by its parabolic graph, which opens either upwards or downwards depending on the sign of the coefficient $$a$$. Quadratic functions are fundamental in understanding convexity, as they often serve as the simplest examples of convex functions when $$a > 0$$ and concave functions when $$a < 0$$.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function is a parabola that can either open upwards if $$a > 0$$ or downwards if $$a < 0$$.
2. Quadratic functions have a unique vertex point that represents either the maximum or minimum value of the function.
3. The axis of symmetry for a quadratic function can be found using the formula $$x = -\frac{b}{2a}$$.
4. The roots or zeros of a quadratic function can be determined using factoring, completing the square, or applying the quadratic formula: $$x = \frac{-b \pm \sqrt{D}}{2a}$$.
5. Quadratic functions play a significant role in optimization problems due to their characteristic shapes, which can illustrate local maxima and minima.

Review Questions

• How does changing the coefficient 'a' in a quadratic function affect its graph?
  • Changing the coefficient 'a' in a quadratic function impacts the direction and width of the parabola. If 'a' is positive, the parabola opens upwards, indicating a minimum point at the vertex. Conversely, if 'a' is negative, it opens downwards, indicating a maximum point. Additionally, larger absolute values of 'a' make the parabola narrower, while smaller absolute values make it wider.
• Explain how the discriminant can be used to determine the number and type of roots for a given quadratic function.
  • The discriminant, given by $$D = b^2 - 4ac$$ for a quadratic function in standard form, reveals crucial information about the roots. If $$D > 0$$, there are two distinct real roots; if $$D = 0$$, there is exactly one real root (or a repeated root); if $$D < 0$$, then there are no real roots but rather two complex roots. This provides valuable insight into solving and analyzing quadratic equations.
• Assess how understanding quadratic functions contributes to broader concepts in optimization and convex analysis.
  • Understanding quadratic functions is vital in optimization because they provide clear examples of convex and concave shapes. In convex analysis, a quadratic function with a positive leading coefficient is convex and has unique properties that simplify finding global minima. This relevance extends beyond just math; in fields like economics and engineering, analyzing systems modeled by quadratic functions allows for efficient decision-making regarding resource allocation and maximizing outputs.

Related terms

Vertex: The vertex is the highest or lowest point on the graph of a quadratic function, depending on whether it opens downward or upward.

Discriminant: The discriminant, denoted as $$D = b^2 - 4ac$$, determines the nature of the roots of a quadratic equation, indicating whether there are two real roots, one real root, or two complex roots.

Standard Form: The standard form of a quadratic function is expressed as $$f(x) = ax^2 + bx + c$$, which allows for easy identification of coefficients and the vertex of the parabola.