Quadratic relations

By Deepak Gogna

Topics

Non-linear relations

Transformation of quadratics

Quadratic relations

Graphing y=(h-k)2+k

factored form- Y=a(x-r)(x-s

special product

multiplying polynomials

common factors

factor quadratic expression of form x2+bx+c

factor quadratic expression of form ax2+bx+c

factor a perfect square trinomial and difference of squares

maxima and minima

solve quadratic equations

graph quadratics using the x-intercept

The quadratic formula

solving word problems using the quadratic formula

What is quadratics?

There are many different ways to write quadratic relations but here are the most common. The first way is Vertex Form, the second way is Factored Form and the third way is standard form.Vertex Form is : y= a(x-h)2 + K

Factored form is : y= a(x-r)(x-s)

Standard Form is : y =ax2 + bx + c

A quadratic relationship is shown by a parabola.

You can tell if its a quadratic by first and second differences.

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Transformation of quadratics

Transformations:


The (-h) moves the vertex of the parabola left or right


The (k) moves the vertex of the parabola up or down


The (a) stretches the parabola and if the


Vertex= (h,k)


Once the vertex of a parabola is indicated the step pattern is used to find the rest of the points.

Graphing a parabola in vertex form

optimal value

Optimal value is the "k" in vertex form. Its the max or min value on a parabola.

complex factoring

The different about complex factoring is that it does not have a coeffiecient of 1 and there are no common factors.

An example is :

4x^2 +6x + 4

common factoring

Common factoring is used when the equation can easily be factored by isolating a common factor outside the bracket

First take out all common factors in equation

Also take out any common variables

Mission is to take out all common factors

simple factoring

when an equation begins with the number 1 in most equations it is a simple trinomial.

Perfect Squares

If a trinomial starts and ends with a square, it might just be a perfect square equation.

A perfect square is a square of a whole number, a number that is squarable.

A perfect square's equation looks like

a^2 + 2ab +b^2

Factoring perfect square trinomials

difference of squares

n an equation, when two squared numbers are involved and one is getting subtracted by the other, it is known as difference of squares.

A difference of squares equation looks like...

a^2 - b^2

Factoring difference of squares

The quadratic formula

Step 1: Find the coefficients a , b and c


Step 2: Plug in the values for a, b, and c into the quadratic formula


Step 3: Simplify


Step 4: Solve for X

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step pattern

The step pattern shows how steep the parabola is. Also it is used to find the next points after the vertex is found. The equation for the step pattern is y=x^2. The pattern goes over 1 then up 1, over 2 and up 4, etc. The "a" can change the steepness, you can continue this pattern to find the next points.

Maxima and Minima

Maxima and minima are the lowest or highest

points on a parabola. When the vertex is the lowest point on the parabola

then it is know as the minimum value. Also If the vertex is at the highest point on

the parabola then it is know as the maximum value.

Solving Quadratic Equations by Factoring - Basic Examples

Multiplying polynomials

Multiplication of Polynomials

Connections

When converting standard form into vertex form you have to use completing of squares

When you are changing standard form into vertex form you have to use factoring

Lastly When you are converting vertex form into standard form you have to use expanding and simplifying

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