Quadratic Functions. Graphing and Finding the roots, zeros, or solutions.

Quadratic? Function? Is there a simple way to figure it out? How is different graphing a parabola using its features from other methods like finding the vertex and then the solutions? ... How many ways may I find the solutions? By factoring?...What if I am having difficulties with factoring? is there another method?

The real world has many examples of a mathematical model to represent paths of projectiles launched from a platform, or a cannon ball going against the enemy troops, or when finding the optimal curvature for a road going in perpendicular direction of a highway. Also this is true when you try to maximize the volume of a prism while minimizing the material that is being used. There are many, many examples that use a quadratic regression when you want to find an equation to do predictions about it. Fantastic lesson that presents quadratic functions (parabola) with all the possible solution paths that we may use to graph it, or to find the solutions. It shows how to find the vertex form of the parabola equation from the graph and from a set of coordinate points. While challenging, it is simply put. Simple! But help yourself! Scroll down the page and review the vocabulary given in that section. Also, a good student highlights important sections and writes marginal notes inside the lesson while studying it. You may accomplish this by using the MARKER TOOLS menu provided in the upper right section of the lesson's screen. After each example you are given a similar problem to work it yourself. Once the problem has been attempted you may see the solution in the following slide.

Lesson's Content


Lesson's Glossary

Axis of Symmetry: A line on which a graph is reflected onto itself.

Completing the Square: Method that finds the constant term in an incomplete perfect square trinomial of a second degree equation to solve it.

Factoring: The process to brake a polynomial down into the product of several factors.

Factors: All whole numbers that are multiplied together to yield another number.

Factored Form: Any polynomial that is written as the product of polynomials of lower degree that may be obtained from the original polynomial.

Function: A relation of the type that has exactly one value in the domain (independent variable) matching a value in the range (dependent variable).

Function notation: A function written with the symbol f(x) instead of y. It is read as f of x.

Quadratic equation: An equation of the form.

ax2 + bx + c = 0 where a, b, and c are real numbers and a is different from zero.

Quadratic formula: If ax2 + bx + c = and a is different from zero then the quadratic formula is given in terms of a, b, and c.

Quadratic function: Any function in the form of

f(x) = ax2 + bx + c where a is different from zero. The graph is a parabola and the largest exponent is 2.

Square Root: If x2 = y, then x is the square root of y. Square root is the opposite of square.

Solution or root: The value that makes an equation a true statement, a root refers particularly to the value of x for which y = 0, this value is also the x-intercept of the graph.

Vertex: The lowest point for a parabola that opens up (minimum); the highest point for a parabola that opens down (maximum).

Vertex form of a quadratic function: The vertex form of a quadratic function is: f(x) = a(x-h)2 + k. The vertex coordinates are (h,k).

Zeros of a function: The solutions for the equation of the function when this equal to 0. The roots, also known as the x-intercepts.


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