Quadratics 101!

By: Harkirat Singh Bassi

Table of Contents

Vocabulary


Linear or Quadratic

Parabolas

Graphing Parabolas Video

Factoring

Types of Forms

All About Vertex Form

The Axis of Symmetry and Optimal Value

Transformations
Zeros & X-intercepts

Step pattern

All About Factored Form

Zeros & X-intercepts

Axis of Symmetry

Optimal Value

All About Standard Form

Zeros

Axis of Symmetry

Optimal Value

Completing the Square to Turn to Vertex Form

Factoring to Turn to Factored Form

  • Common
  • Simple Trinomial
  • Complex Trinomial
  • Perfect Squares
  • Difference of Squares

Word Problems

Making Connections

Reflection

Vocabulary

Non-linear relation:

a relationship between two variables that does not follow a straight line when graphed.

Curve of best fit:

a smooth curve drawn to approximate the general path or trend in a scatter plot.

Parabola:

the graph of a quadratic relation, which is U-shaped and symmetrical

Vertex:

The point on a parabola where the curve changes direction. The maximum point if the parabola opens down. The minimum point if the parabola opens up.

Axis of symmetry:

the line that divides a figure into two congruent parts.

Zero:

a value of x for which a relation has a value of 0. Corresponds to an x-intercept of the graph of the relation.

x-intercept:

the x-coordinate of the point where a line or curve crosses the x-axis. At this point y=0

Perfect square trinomial:

a trinomial that is the result of squaring a binomial.

Difference of squares:

an expression that involves the subtraction of two squares.

Linear or Quadratic

How can you tell if a relation is linear or non linear?


A relation is quadratic if the degree of the equation is 2, in other words the highest exponent in the equation is 2.


Another way to find out is by checking the finite differences. If the first differences are constant than its linear. If the first differences are not constant and the second differences are then that mean the relation is quadratic.

Big image

Parabolas

  • Parabolas can open up or down
  • The zero of a parabola is where the graph crosses the x-axis
  • "Zeros" can also be called "x-intercepts" or "roots"
  • The axis of symmetry divides the parabola into two halves
  • The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
  • The optimal value is the y co-ordinate of the vertex
  • The y-intercept of a parabola is where the graph crosses the y-axis

To graph parabolas you can use Desmos. It is a quick substitute to drawing a parabola onto graph paper.

Big image

Factoring

In order to really understand quadratics, how to make parabolas and how to convert forms you must know how to common factor properly. Factoring is a big part of quadratics. The video below will help you understand common factoring better.
Common Factors

Types of Forms

Vertex form - a(x-h)²+k

Factored form - a(x-r)(x-s)
Standard form - ax²+bx+c

Vertex Form - y=a(x-h)^2+k

Axis of Symmetry

The axis of symmetry can be found in an equation in the vertex form when looking at the "h" value. (Shown as x=-1(h))


***The value of -1(h) is the axis of symmetry. Opposite Signs***


Examples:

Equation:

1. y=2(x-6)^2+7

2. y=5(x+4)^2+21


AOS:

1. x=6

2. x=-4

Big image

Optimal Value

The optimal value is the "k" in vertex form. It is the maximum or minimum value on a parabola. (Shown as y=k)


Examples:

Equation:

1. y=2(x-6)^2+7

2. y=5(x+4)^2-21


Optimal Value:

1. y=7

2. y=-21

Big image

Vertex

The vertex of the parabola can be found by taking the coordinates of the axis of symmetry and optimal value. (Shown as (x,y) )

Example:

Axis of Symmetry

x=3

Optimal Value

y=5

Vertex Coordinates

(3,5)

Big image

Transformations

Horizontal Transformation - Moves the parabola left and right

-If the h value is less than zero, then it will move to the right

-if it is more than zero, then it will move left.


Vertical Transformation - Moves the parabola up and down

-If the k value is greater than 0, the parabola will move up

-If the k value is less than 0, the parabola will move down


Vertical Stretch

  • Increasing the value of a will vertically expand the graph (make the arms move closer together

  • Decreasing the value of a will vertically compress the graph (make the arms spread out wider).


Reflection - Determines whether the direction of opening is upwards or downwards

  • If the "a" value is greater than 0, direction of opening is upwards

  • If the "a" value is less than 0, direction of opening is downwards

Step Pattern


The step pattern shows how steep the parabola gets and use it to find next points after vertex is found. The step pattern is y=x^2. So it is over 1 and up 1, over 2 and up 4, etc. The "a" can change the steepness, have to multiply the "a" by the 1 to find next point.

Big image

Zeroes


To find the zeroes in vertex form, you need to sub in y=0 and isolate for x.

Factored Form

Factored Form of a quadratic looks like y=a(x-s)(x-t).


Zeroes

The x-intercepts are at (s, 0) and (t, 0) and they will be opposite. for example, if "s" was 2 and "t" was 3 in the equation ((y=a(x-2)(x-3)), then the zeroes would be at (2,0) and (3,0) because they are opposite of "-2" and "-3" .


Zeros & X-intercepts

y = ( x - 3 ) ( x - 4 )

x - 3 = 0 x - 4 = 0

x = 3 x = 4


Axis Of Symmetry

To find the axis of symmetry, need to sub in x-intercepts into x=s+t/2.


Axis of Symmetry

x = 3 + 4 / 2

x = 3.5


Optimal Value

To get the optimal value, need to get the "x" from AOS and sub into the equation, y=a(x-s)(x-t), this will give the y value and then use the x and y coordinates to get vertex.


Optimal Value

y = ( x - 3 ) ( x - 4 )

y = ( 3.5 - 3 ) ( 3.5 - 4 )

y = 12.25 - 14 - 10.5 -12

y = -24.25


Expanding and Simplifying

When expanding from factored form, first the "a" will be multiplied to the "x" and "s" in first bracket and then "x" from the first bracket must be multiplied to other "x" in second bracket. Then the "x" from first bracket is multiplied to "t." After, the "s" is multiplied with "x" from second paragraph and then "t." At the end collect like terms.

3.5 Graphing from Factored Form

Quadratic Formula

The quadratic formula is very easy once you know what the formula is. All you have to do is sub the value of a, b and c. The formula is -b±√b^2-4ac/2a and with this formula you can find the two x values because a parabola can have two x-intercepts. Also be careful when you are doing a word problem because sometimes the value of x should not be negative, like for a area question the length or width can't be negative.
Big image
Quadratic Formula

Standard Form

Standard Form of a quadratic looks like y=ax^2+bx+c


Zeroes

To find the zeroes of standard form, will have to use quadratic formula.


Axis Of Symmetry

After the x intercepts are found using the quadratics formula, then add the two together and divide the result by two and it will give the "x" value.


Optimal Value

To get the optimal value in standard form, need to sub in "x" from AOS into the equation and solve. This will give the y coordinate for the vertex.

Big image
Below is the step by step process of completing a square. This is used to convert standard form to vertex form.
Big image
Factoring


Next we will factor to convert from standard form to factored form.

Common Factoring

Common factoring is finding what is common between the two equations, then divide the whole thing by what is common. Then with what is left will go in bracket and what was used to divide, will now be used to multiply with what is in brackets.

Big image

Simple Trinomial Factoring

A simple trinomial is when you have three different terms, but the "a" value is equal to one in standard form equation, y=ax^2+bx+c. This will become factored form at the end. Two numbers must multiply to "c" and add to "b".

Big image

Complex Trinomial Factoring

Use complex trinomial factoring when the "a" value is higher than one in standard form. Instead of putting an "x" at the beginning, put what are the multiples of the "a". For example if it was 3x^2+1x+1, then we would do (3x+1)(x+1) because factors of 3 are 1 and 3 so we use those if it has more then guess with both that mutiply to the number.

Big image

Perfect Squares Factoring

Check if you can factor "a" and "c" value in standard form y=ax^2+bx+c. If can factor those two terms, then can factor a perfect square. When factoring perfect square, it looks like this: (a+b)^2 which can also be a^2+ 2ab + b^2.


Difference Of Squares Factoring

Difference of square is when there are squares at beginning and end but they have no middle term. After factored, one bracket should be adding two terms and other should be subtracting.

Big image

Word Problems

The First Example
Big image
Second Example
Big image
Third Example
Big image

REFLECTION

Through out the units of quadratics I learned a lot. I learned how quadratics could be used in real life situations. For example when you are trying to make maximum amount of profit by how much do you need to decrease of increase the price of the object to make maximum profit. Quadratics also related to physics because of the projectile equation. I have realized that quadratics is not only just to learn and forget, it can be used in many real life situations. I learned how mathematicians derived the quadratic formula by using completing the square on the basic standard form, y=ax^2+bx+c. I believe I recieved a thorough knowlede about the quadreatics unit. After finishing all three chapters I like quadratics because of the shortcuts and because now I have a variety of ways to solve a quadratic equation.
Big image
Big image