Quadratics

Everything You Need to know About Quadratics

Learning Goals for Quadratic Unit

  • I could use a Table of Values to create a graph
  • I could Identify the difference parts of the Quadratic Equation
  • I can graph the base graph of a quadratic function with no technology

Introduction of the Quadratic Unit

What is a Parabola?

A Parabola is a curve that is at equal distance from the fixed point and the fixed line (Line of symmetry)

Vertex Form

Overview of Vertex Form

The formula of a quadratic in a vertex form is:

y=a(x-h)2 + K (Note that the 2 is a squared)


The value of "A" in this formula, determines whether the parabola is stretched (Narrow) or is compress (Wider). As the value of "A" increase, the more narrow it gets. As value of "A" decreases, the parabola gets wider. It also tells you the the direction of opening of the parabola. If the "A" value is a negative, the direction of opening is down and vice versa.


The "H" in this equation determines whether the parabola will move to the right on the graph or left in the graph. This is more tricky than "A" and "K" because this is backwards. If the value of "H" is -2, the parabola moves right on the graph. But if it is a +2 in the equation, it moves to the left of the graph.


The "K" in this formula, determines whether the parabola will move up or down on the graph. If the "K" is 2, it moves 2 spaces up and if the value of "K" -3, the parabola moves 3 spaces down.


The value "H" ( X) and "K" (Y) are your vertex. But make sure you change the value of "H" to the opposite (-2 to 2).

How to Solve A Quadratic Word Problem?

To solve a word problem involving a parabola. You must know the terms, so you know the what the question is asking for.


The Maximum Height is the "Y" intercept of the vertex.


Time Reached Maximum Height is the "X" intercept of the vertex.


To find the Initial Height, you must substitute the "X" with zero. To find your answer.


To find the Height of the object at a specific time, you substitute the "X" with the specific time.

Word Problem (Vertex Form)

Reyad threw a baseball to his friend, Jose. The equation that he threw the ball was

y= -(x-4)2 + 4. Where y equals height in meters and x equals time in seconds.


A) At what height did the ball reach the maximum height? and at what time?


y= - (x - 4)2 + 4 (x,y) to (h,k) to (4,4)


The h and k in this equation are the answer.

Therefore the maximum height in which the ball reached was 4 meters at 4 seconds.

Factoring Form

Learning Goals

  • I could factor using different methods
  • I need to know what a,r and s in the Factoring Equation are so that you know how to use it right

Overview of Factoring Form

The equation that is used for Factoring is:

y=a(x-r)(x-s)


The "A" in this equations tells you the direction of the opening of the parabola. If "A" is a negative in the equation, the direction of opening is downwards. Also, if "A" is a positive, the direction of opening, is upwards of the parabola. "A" also determines the shape of the parabola (compress or stretched)


The "R" and "S" in this equation are the x-intercepts of the equation.


Axis of Symmetry:


To find the A.O.S, you sub the x value in the equation to find the optimal value.

x= (r+s/2) is the equation you substitute the x value to find the optimal value.


To find the y-intercept, you set x as 0. With this you solve for the y-intercept.

Expanding and Simplifying (Factored Form)

This image shows how to expand the coordinates and simplify it.


Example:

(3x + 2)(2x - 5)

= 6x2 - 15x + 4x - 10

= 6x2 - 11x -10

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How to Factor any Quadratic Equation

Monomial Common Factor (GCF)

To factor a "Monomial", you must find the GCF of the equation and divide it from the equation, to cancel it out. You put the GCF outside the bracket and bracket the remainder of the equation.


Example:

3x + 6y

= 3(x + 2y)

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Factoring Common Monomial Factor

Binomial Factoring (GCF)

If the equation is a binomials, treat it as two different monomial. You do the same thing as factoring "Monomial Common Factoring" but you do it twice in a Binomial Factoring.
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Factoring by Grouping

To factor by grouping, you bracket two set of the numbers. You then find the GCF of each bracket. Both numbers in the bracket should be the same. You take the that bracket that has same and join the two of the GCF to fully factor it.
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Factoring by Grouping - Ex 1

Factoring Simple and Complex Trinomials

To factor "Simple Trionomial" you must find a two numbers that when are multiplied equal a + b but when you add the two numbers together it must equal the "B". After you find the two numbers, you bracket two terms within the equation. You then find the GCF of both terms and then add the GCF together to make one term and the term it must be the same.
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Example 1: Factoring trinomials with a common factor | Algebra II | Khan Academy

Perfect Square Trinomials

A perfect square trinomials are basically when the "A" and "B" have to the ability to be squared rooted.


Such as: 25x2 + 200x + 4

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Quadractics on a Graph

This picture shows a parabola on a graph. It also shows all the different part of the parabola, on a graph.

Differences of Squares

Differences squares are numbers that have the ability to be square rooted. All you do is square root both variables (A and B).


An example of Difference of Square is:

(3x - 8)(3x + 8)

Word Problem (Factored Form)

The sum of three consecutive odd integers is 231. What are those integers?


Let x rep smallest integer

Let x + 2 rep 2nd integer

Let x + 4 rep 3rd integer


x + x + 2 + x +4 = 231

3x + 6 = 231

3x = 231 - 6

3x = 225

3x = 225/2

x = 75


Therefore the three integers are 75, 77 and 79

Standard Form

Overview of Standard Form

The equation that is used for Standard form is:

y=ax2 + bx + c


The value of "A" gives you the shape and the direction of opening of the parabola. The value of "C" is the y-intercept of the parabola.

Learning Goals

  • I could graph the parabola using the Quadratic Formula
  • I could find the x-intercepts using the Quadratic Formula

Quadractic Formula

The Quadratic Formula is:
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To use this equation, you must substitute the unknown using the "Standard Form" equation.


An example:

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Completing the Squares

To complete the square you must place brackets between ax2 and bx.

Like so:

(ax2 + bx) + c


If the a has a value more than one you must divide it out of the equation to isolate the x.

Then you have to divide b by 2 and then square it, so you can add the number within the bracket.

You then subtract it in the bracket and then subtract it out of the bracket to balance the equation.

You then take the number you square it by and then substitute it into the vertex form equation.


Example:

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Word Problem

Question:

Reyad is practicing her circus act by flying out a cannon. The equation that represents her flight is h= -5t2 + 20t + 2. With her height represents as h, in meters and t represents time in seconds.

A)What is the maximum height that Ms.Dhaliwal reaches?

B)At what time did she reach the maximum height?

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Reflection

The quadratic unit was a tricky unit. It had three different parts; vertex form, factored form and standard form. The equation I like to use the most is the standard form equation. The reason for this is because I find it easier to find the solution, because since the quadratic formula has pretty long, you find the solution with less effort.

Factoring connects to graphing by when you factor you find the expression for the x-intercepts, which you can use to find the x-intercepts of the parabola. This connects to graphing by you use factoring to find the vertex of the parabola, which you can use to plot on the graph.