quadratic function

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.

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
  • Linear and quadratic functions make lines and parabola, respectively, when graphed.
  • A quadratic function, in mathematics, is a polynomial function of the form: $f(x)=ax^2+bx+c, a \ne 0$.
  • The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis .
  • If the quadratic function is set equal to zero, then the result is a quadratic equation.

• 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 $f(x) = x^2 - x - 2$.

• 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