Quadratic Function (Polynomial), Equation & Formula

Types of Functions >

Contents:

1. Quadratic Function
   • Domain and Range of a Quadratic Function
2. What is a Quadratic Term?
3. Quadratic Equation vs. Quadratic Function
4. How to Use the Quadratic Formula in Calculus
5. Parabolic Function Graph

1. What is a Quadratic Function?

A quadratic function (also called a quadratic, a quadratic polynomial, or a polynomial of degree 2) is special type of polynomial function where the highest-degree term is second degree.

The general form of a quadratic function is:
f(x) = ax² + bx + c (or y = ax² + bx + c) ,
where a, b and c are all real numbers and a cannot be equal to 0.

The graph of a quadratic function is a parabola, a 2-dimensional curve that looks like either a cup(∪) or a cap(∩).

The quadratic function y = x² – x – 2 is plotted below:

The “a” variable of the quadratic function tells you whether a parabola opens up (more formally called concave up) or opens down (called concave down).  In the graph above the variable x²  is positive so that parabola opens up.  If the variable x²  were negative, like -3x², the parabola would open down.  A parabola that opens up has a vertex that is a minimum point.  A parabola that opens down has a vertex that is a maximum point.  Quadratic functions are useful when trying to solve problems involving quantities with unknown variables.

Identifying a Quadratic Function

The quadratic function is an example of a second-degree polynomial.  What this means is that the highest degree that a variable in the function can have is 2.  One way to remember this is to keep in mind that the term quadratic comes from the word quadrate which means square (we are looking for a function where squared is the highest variable).

Examples

• f(x) = x²  is a quadratic function because the degree of variable x is 2.
• f(x) = 2x² + 5x + 3 is a quadratic because the highest degree of all of the variables is 2.  Note that the degree of variable 5x is 1.
• f(x) = x³ is not a quadratic. The degree of the x variable is 3.  Again, the highest degree of an x variable in a quadratic function is 2.
• f(x) = x⁵ – 6x².  This one is a bit more tricky since we have an x variable with a degree of 2.  However, the highest degree in the function is a 5, which violates the general rule above.  Thus, this function is not a quadratic function.

Domain and Range of a Quadratic Function

A quadratic function is a type of polynomial function. You can use the following flow chart to find the domain and range:

What is a Quadratic Term?

A quadratic function or equation has the form f(x) = ax² + bx + c. It contains three terms:

• ax² = the quadratic term (a is the leading coefficient). This term is always raised to 2, so is sometimes called the squared term.
• bx = the linear term,
• c = the constant term.

For example, for the function f(x) = 5x² + 4x – 2, the quadratic term is 5x². In the expression f(x) = x² + 2, it’s x². The sign of the leading coefficient affects the shape of the parabola:

• A positive coefficient will result in a “U” shape,
• A negative coefficient will result in a “∩” shape.

It’s called a quadratic term because it’s the term that makes the expression a quadratic function: take away the “ax²” and you’re left with a linear function, bx + c.

Quadratic Term Example

Projectile motion can be described by, in part, a quadratic term. For example, the following graph shows the path of a baseball:

The expression needed to fully explain the baseball’s path has three parts:

• The quadratic term represents the ball’s motion due to gravity. That’s what gives the path the distinctive quadratic (i.e. humped) shape: acceleration slows down as gravity pulls on the ball. Eventually the ball reaches its peak height, then gravity slows the ball down further until it eventually hits the ground.
• The linear term represents the vertical (upwards) motion of the ball after it is hit by the bat. This part of the expression would be a straight line, except that gravity bends it downwards.
• The constant term represents the ball’s initial height when hit. For a baseball, this would be roughly 2-3 ft, depending on the height of the player.

2. Quadratic Function vs. Quadratic Equation

In calculus, we’re mostly concerned with functions.

An equation is simply an expression with two equal terms. For example:
y = ax² + bx + c
If the quadratic equation meets the requirement for functions (that each input is matched to at most one output), then it’s called a quadratic function. Usually, you would give that function a name:
f(x) = ax² + bx + c.
That naming convention, where “f” is the function relationship, means that it’s easier to set quadratic functions and quadratic equations apart.

Solving a Quadratic Equation

A quadratic equation always has two solutions, as it is a second-order polynomial. These two solutions can be both either real numbers or complex numbers.

You can find the roots (or zeros) by factoring or by completing the square. Sometimes it may be practical to graph the equation; Often the easiest way to find the roots is by using the quadratic formula, which, when you plug in the a, b, and c of your particular quadratic equation, will give you the roots.

This formula can be derived by completing the square of the generalized equation ax² + bx + c = 0. Completing the square is where you rewrite a polynomial as a quantity raised to the power of “2”.

Quadratic equations will often come up in algebra, and the quadratic formula is worth memorizing. They also have many applications in geometry, engineering, and biology.

3. How to Use the Quadratic Formula in Calculus

The quadratic formula is a way to find the solution for any polynomial in the form ax² + bx + c = 0. You’ll need to use the quadratic formula to find the solutions for polynomials in many places; for example, you can use solutions for polynomials to find total distance for velocity equations. It works when factoring doesn’t, or when factoring is too complicated. In fact, the formula will always give you a solution, while factoring sometimes works and sometimes doesn’t.

The quadratic formula is stated as:
For any function of the form ax² + bx + c = 0, the value of x is given by:


“a”, “b” and c are just numbers, or numerical coefficients. The formula is derived from completing the square.

Quadratic Formula Example

Example problem: Solve x² + 3x + 4 using the quadratic formula.

Step 1: Identify a, b and c in your function.
We know that the quadratic formula is: ax² + bx + c = 0. So, in this example, a = 1, b = 3 and c = 4. Note that there isn’t a number before x², so we put a “1.” That’s because 1 * x² = x².

Step 2: Insert a, b, and c into the quadratic formula: