quadratic function

(noun)

A function of degree two.

Related Terms

Examples of quadratic function in the following topics:

  • What is a Quadratic Function?

    • A quadratic function is of the general form:
    • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
    • All quadratic functions both increase and decrease.
    • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.
    • Quadratic functions can be expressed in many different forms.

  • Parts of a Parabola

    • The graph of a quadratic function is a parabola, and its parts provide valuable information about the function.
    • The graph of a quadratic function is a U-shaped curve called a parabola. 
    • In graphs of quadratic functions, the sign on the coefficient $a$ affects whether the graph opens up or down.
    • There cannot be more than one such point, for the graph of a quadratic function.
    • Recall that if the quadratic function is set equal to zero, then the result is a quadratic equation.

  • A Graphical Interpretation of Quadratic Solutions

    • The roots of a quadratic function can be found algebraically or graphically.
    • Consider the quadratic function that is graphed below.
    • Find the roots of the quadratic function $f(x) = x^2 - 4x + 4$.
    • Therefore, there are no real roots for the given quadratic function.
    • Graph of the quadratic function $f(x) = x^2 - x - 2$

  • The Discriminant

    • The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots.
    • Because adding and subtracting a positive number will result in different values, a positive discriminant results in two distinct solutions, and two distinct roots of the quadratic function.
    • Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
    • This means the square root itself is an imaginary number, so the roots of the quadratic function are distinct and not real.
    • Graph of a polynomial with the quadratic function f(x) = x^2 - x - 2.

  • Financial Applications of Quadratic Functions

    • For problems involving quadratics in finance, it is useful to graph the equation.
    • From these, one can easily find critical values of the function by inspection.
    • The method of graphing a function to determine general properties can be used to solve financial problems.Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts.
    • If one needs to determine several values on a quadratic function, glancing at a graph is quicker than calculating several points.
    • Suppose this models a profit function $f(x)$ in dollars that a company earns as a function of $x$ number of products of a given type that are sold, and is valid for values of $x$ greater than or equal to $0$ and less than or equal to $500$.

  • Graphing Quadratic Equations In Standard Form

    • A quadratic function is a polynomial function of the form $y=ax^2+bx+c$.
    • Regardless of the format, the graph of a quadratic function is a parabola.
    • Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
    • The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
    • Explain the meanings of the constants $a$, $b$, and $c$ for a quadratic equation in standard form

  • The Quadratic Formula

    • The first criterion must be satisfied to use the quadratic formula because conceptually, the formula gives the values of $x$ where the quadratic function $f(x) = ax^2+bx+c = 0$; the roots of the quadratic function.
    • Conceptually, this makes sense because if $a=0$, then the function $f(x) = ax^2 + bx+c$ is not quadratic, but linear!
    • Suppose we want to find the roots of the following quadratic function:
    • First, we need to set the function equal to zero, as the roots are where the function equals zero.
    • Solve for the roots of a quadratic function by using the quadratic formula

  • Completing the Square

    • Completing the square is a method for solving quadratic equations, and involves putting the quadratic in the form $0=a(x-h)^2 + k$.
    • Along with factoring and using the quadratic formula, completing the square is a common method for solving quadratic equations.  
    • The value of $k$ is meant to adjust the function to compensate for the difference between the expanded form of $a(x-h)^2$ and the general quadratic function $ax^2+bx+c$.  
    • This quadratic is not a perfect square.  
    • Solve for the zeros of a quadratic function by completing the square

  • Standard Form and Completing the Square

    • In algebra, parabolas are frequently encountered as graphs of quadratic functions, such as:
    • Completing the square may be used to solve any quadratic equation.
    • This can be applied to any quadratic equation.
    • Graph with the quadratic equation .
    • The graph of this quadratic equation is a parabola with x-intercepts at -1 and -5.

  • Other Equations in Quadratic Form

    • Many equations with no odd-degree terms can be reduced to quadratics and solved with the same methods as quadratics.
    • If a substitution can be made such that the higher order polynomial takes the form of a quadratic, any method for solving a quadratic equation can be applied.  
    • For example, if a quartic equation is biquadratic—that is, it includes no terms of an odd-degree— there is a quick way to find the zeroes of the quartic function by reducing it into a quadratic form.  
    • Consider a quadratic function with no odd-degree terms which has the form:
    • Use the quadratic formula to solve any equation in quadratic form