Quadratic function

Definition

A quadratic function is a type of polynomial function that can be expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $a$, $b$, and $c$ are constants and $a \neq 0$. This function forms a parabola when graphed, which can open either upward or downward depending on the sign of the coefficient $a$. Quadratic functions are essential in various applications, including physics, engineering, and finance, as they can model real-world phenomena like projectile motion and profit optimization.

5 Must Know Facts For Your Next Test

1. The graph of a quadratic function is always a parabola, which can be either concave up (if $a > 0$) or concave down (if $a < 0$).
2. The vertex of the parabola provides crucial information about the function, including its maximum or minimum value.
3. Quadratic functions can intersect the x-axis at zero, one, or two points based on the value of the discriminant.
4. Real-world applications of quadratic functions include modeling projectile motion in physics and optimizing areas in geometry.
5. Completing the square is a method used to rewrite a quadratic function in vertex form, making it easier to analyze its properties.

Review Questions

• How can you determine whether a quadratic function opens upwards or downwards?
  • To determine if a quadratic function opens upwards or downwards, examine the leading coefficient $a$ in its standard form $$f(x) = ax^2 + bx + c$$. If $a > 0$, the parabola opens upwards and has a minimum vertex. If $a < 0$, the parabola opens downwards and has a maximum vertex. This property is crucial for understanding the behavior of the function and finding its extremum.
• Describe how to use the discriminant to analyze the roots of a quadratic function.
  • The discriminant, calculated as $$D = b^2 - 4ac$$ from the quadratic equation $$ax^2 + bx + c = 0$$, provides insight into the nature of the roots. If $D > 0$, there are two distinct real roots; if $D = 0$, there is one real root (the vertex touches the x-axis); and if $D < 0$, there are no real roots, indicating that the parabola does not intersect the x-axis. Understanding this helps in predicting solutions to quadratic equations.
• Evaluate the significance of vertex form in understanding quadratic functions and their applications.
  • Vertex form, expressed as $$f(x) = a(x-h)^2 + k$$, where $(h,k)$ is the vertex, is significant because it highlights key features of a quadratic function, such as its maximum or minimum value and its axis of symmetry. This form is particularly useful for graphing and analyzing how changes in parameters affect the parabola's position and shape. In applications like profit maximization or projectile motion, knowing the vertex allows for quick identification of optimal outcomes.

Related terms

vertex:

The highest or lowest point on the graph of a parabola, which represents the maximum or minimum value of a quadratic function.

discriminant: A value calculated from the coefficients of a quadratic equation that determines the nature of the roots; it is given by $$D = b^2 - 4ac$$.

axis of symmetry:

A vertical line that divides the parabola into two symmetrical halves, given by the equation $$x = -\frac{b}{2a}$$ for a quadratic function.