Difference between revisions of "Quadratic Functions"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
Line 43: Line 43:
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [https://en.wikibooks.org/wiki/Algebra/Quadratic_functions Quadratic functions, Wikibooks: Algebra] under a CC BY-SA license
 
* [https://en.wikibooks.org/wiki/Algebra/Quadratic_functions Quadratic functions, Wikibooks: Algebra] under a CC BY-SA license
 +
* [https://en.wikibooks.org/wiki/Algebra/Quadratic_Equation Quadratic Equation, Wikibooks: Algebra] under a CC BY-SA license

Revision as of 13:08, 21 October 2021

A Quadratic function is a polynomial where the highest power is two. The basic form of this function is

where, ax² is the quadratic term, bx is the linear term and c is the independent term or "constant", which does not depend on the variable, x. The letters a and b are called "coefficients", a being the "leading coefficient". The standard form is F(X) = ax² + bx + c. The x-intercepts of the function are:

OR

The independent term is also the y-coordinate of the point of intersection with the y-axis (when X=0, F(X)=C).

A quadratic function has a "vertex" or "turning point", which is the point where the function has either a maximum or minimum value. If a is greater than zero, then there will be a minimum and the curve will be concave. If a is less than zero, then there will be a maximum and the curve will be convex. If a = 0, then we have a linear function rather than a quadratic function.The x-coordinate of the vertex is The y-coordinate of the vertex is

The general form of a quadratic equation is actually F(X) = ax² + bxy + cy² + dx + ey + f = 0, which can take many shapes including circles, ellipses and parabolas, but in most Western high schools, quadratic equation refers only to those of the form F(X) = ax² + bx + c, which forms a parabola.

Deriving the Quadratic Equation

The solutions to the general-form quadratic function can be given by a simple equation called the quadratic equation. To solve this equation, recall the completed square form of the quadratic equation derived in the previous section:

In this case, since we're looking for the root of this function. To solve, first subtract c and divide by a:

Take the (plus and minus) square root of both sides to obtain:

Subtracting from both sides:

This is the solution but it's in an inconvenient form. Let's rationalize the denominator of the square root:

Now, adding the fractions, the final version of the quadratic formula is:

This formula is very useful, and it is suggested that the students memorize it as soon as they can.

Resources

Licensing

Content obtained and/or adapted from: