Quadratic function

Definition

A quadratic function is a type of polynomial function represented by the formula $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a$$ is not equal to zero. Quadratic functions create parabolic graphs that can open either upward or downward, depending on the sign of the leading coefficient, $$a$$. The shape of these graphs, along with their vertex and axis of symmetry, reveals important characteristics related to optimization and convexity.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function is a parabola that opens upwards if $$a > 0$$ and downwards if $$a < 0$$.
2. The vertex form of a quadratic function is given by $$f(x) = a(x - h)^2 + k$$, where (h, k) represents the vertex coordinates.
3. Quadratic functions are always continuous and differentiable over all real numbers.
4. The second derivative of a quadratic function is constant, which indicates its concavity and helps determine whether it's convex or concave.
5. The solutions to the equation $$ax^2 + bx + c = 0$$ are found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$.

Review Questions

• How do you determine whether a quadratic function is convex or concave based on its leading coefficient?
  • To determine if a quadratic function is convex or concave, look at the leading coefficient, $$a$$. If $$a > 0$$, the parabola opens upwards, making it a convex function. Conversely, if $$a < 0$$, the parabola opens downwards, indicating it is concave. This distinction is crucial when analyzing optimization problems related to the quadratic function.
• What role does the vertex play in understanding the characteristics of a quadratic function's graph?
  • The vertex of a quadratic function serves as the point where the graph changes direction and is either the maximum or minimum point. In optimization problems, finding the vertex allows us to determine optimal values for specific applications. The coordinates of the vertex can be calculated from either the standard form of the function or by converting it into vertex form.
• Evaluate how understanding quadratic functions contributes to solving real-world optimization problems.
  • Understanding quadratic functions is essential for solving real-world optimization problems because they model scenarios where relationships are non-linear. For instance, businesses often use quadratic equations to maximize profit or minimize costs by analyzing revenue functions. By identifying critical points like the vertex and using properties like convexity, we can efficiently find optimal solutions in various practical situations.

Related terms

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

Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves, passing through the vertex.

Convex Function: A function is convex if its graph lies above any chord connecting two points on the graph; for quadratic functions, this occurs when the leading coefficient is positive.