QUADRATIC RELATIONSHIPS

By Raghav Sharma

UNIT 1: Intro into Quadratics

UNIT SUMMARY


  • Parts of a Parabola
  • First and Second Differences
  • Vertex Form
  • Word Problems Using Vertex Form
  • Graphing Transformations
  • Mapping Notation

LEARNING GOALS

Learn the parts of a parabola

How calculate first and second differences

Know the vertex form

Learn how to solve word problems using the vertex form

Know all the transformations

Learn mapping notations

Features of a Parabola

Direction of Opening: Can either be Up or Down depending on the position of the vertex relative to the zeroes (if the vertex is lower than the zeroes on a graph, the direction of opening is up. If the vertex higher than the zeroes, the direction of opening is down).

Vertex: The highest or lowest point on a parabola (written as a coordinate pair)

Axis of symmetry: The vertical half point that separates the parabola into 2 equal parts (written as only a number on the x-axis).

Optimal Value: Can either be Minimum or Maximum value. If parabola opens up, there will be a minimum value, and if the parabola opens down, there will be a maximum value (written as only a number on the y-axis)

Zeroes: Also known as roots, are points on the graph where the parabola touches or crosses the x-axis.

Big image

FIRST AND SECOND DIFFERENCES

The method of first and second differences is used to determine if an equation is linear or quadratic. If the first differences between the y-values are all the same, the relation is linear. If the first differences between the y-values are all different, but the second differences are all the same, the relation is quadratic. If neither the first or second differences are equal, the relation is neither linear or quadratic. The following is an example of a quadratic relationship
Big image

VERTEX FORM

The basic vertex form quadratic equation is written as y = a ( x - h ) ^ 2 + k.


1. The a value is the parabola's vertical stretch

- If a is a negative number, the parabola is flipped (opens downwards) or in other words, there is a reflection in the x-axis

2. The h value is the parabola's horizontal translation

- If h = 3, the parabola's axis of symmetry (the middle point) would be at -3 on the x-axis

- If h = -7, the parabola's axis of symmetry would be at 7 on the x-axis


3. The k value is the parabola's vertical translation

- If k = 3, the parabola's optimal value would be at 3 on the y-axis

- If k = -10, the parabola's optimal value would be at -10 on the y-axis

4. By looking at both the h and k value, you can combine both of them to determine the vertex of the parabola

-If the h value is -5 and the k value is 9, the vertex is (5, 9)

Big image

GRAPHING USING VERTEX FORM

With the basic equation y = a ( x - h ) ^ 2 + k , we can plot a parabola using just the information from the equation.


1. As mentioned above, the h value, which is always opposite (negative value is positive on the graph and vice versa) tell us the x value of the vertex


2. The k value, which is NOT opposite, tell us the y value of the vertex

3. The a value will tell you the parabola's vertical stretch. If it is negative, the parabola opens downwards. To use the vertical stretch, you must use the step pattern (one left/right and one up/down. two left/right and four up/down) and multiply the number on the up/down. (e.g. A vertical stretch of 2 is one left/right and two up. two left/right and eight up)



  • To go from vertex form to the equation, first find the vertex's x and y value.
  • The x value is the h. Make sure to make it is negative if it's positive and vice versa
  • The y value is the k. This one is not opposite
  • To find the a value, count once right or left of the vertex and count how many times you have to go up or down until you reach a point where the parabola intersects with both the yand x axis lines. This number is your a value. Your a value will be negative if the parabola opens downwards
  • Finally, input the values in the basic equation and you have the answer



Questions:

y= -3(x+2)^2+4

Quick Way of Graphing a Quadratic Function in Vertex Form

GRAPHING TRANSFORMATIONS

The transformed function f(x) = a(x-k) ^2 + k is the vertex form of the quadratic equation.

When graphing the vertex form there are 4 important things to look for in the given equation:

  1. Vertical Stretch/Compression
  2. Reflection in the x-axis
  3. Horizontal Translation
  4. Vertical Translation


1. Vertical Stretch/Compression

The vertical stretch/compression in a quadratic equation determines whether the parabola contains wide curve or a narrow curve. When a > 1 the parabola has a vertical stretch. When a negative sign is placed in front of it, the rule still applies however it is a < -1. When

0 < a > 1 the parabola has a vertical compression. Just like a stretch, when a negative sign is placed in front of it, the rule still applies but between 0 and -1.


2. Reflection in the x- axis

If the a value given in the quadratic equation has a negative sign in front of it the parabola is reflected into the x-axis meaning it will have a downwards opening. On the other hand, if the a value in the given equation has a positive sign in front of it the parabola will have an upwards opening.


3. Horizontal Translation

The horizontal translation is the variable h in the vertex form. The h value determines where the x in the vertex lies. If the h value is written as a negative in the equation it will be placed on the graph as a positive, and if the h value is written as a positive in the equation it will be placed on the graph as a positive. When h is a negative value the translation on the graph is to the left, and when h is a positive value the translation on the graph is to the right.


Example: y = -1 (x-1)^2 + 1


  • The h value in the equation is -1, meaning on the graph it will be placed as +1
  • This is a horizontal translation 1 unit right


4. Vertical Translation


The vertical translation is the k value in the vertex form. The k value determines where the y in the vertex lies. When the k is a positive the y value moves up and when it is a negative the y value moves down. For example, if the k value is positive 7, the y value in the vertex would be positive 7 and this would be a vertical translation 7 units up.


Example: y = -1 (x-1)^2+1

This is a vertical translation 1 unit up


The Example below shows how the vertical translation effects the placing of a parabola on a graph.

MAPPING NOTATION

Mapping notation is one of the many methods to accurately graph a quadratic equation. In order to learn mapping notation, you must know the key coordinates of the basic quadratic relation y = x². This is the simplest parabola with no transformations applied.

Now, for example, if we were to describe the transformations for the parabola



y = 4 (x - 6)² - 8, they would be:

  • parabola opens upwards
  • there is a vertical stretch by a factor of 4
  • there is a horizontal shift to the right by 6 units
  • there is a vertical shift down by 8 units
  • vertex = (6, -8)
Big image
Through the method of mapping notation, we will see how these translations will affect the coordinates of the key coordinates of the graph y = x².


What really happens to the key coordinates of y = x² to become y = 4 (x - 6)² - 8 :

(x, y) ⟶ (x + 6 , 4y - 8)


MAPPING FORMULA

n general, to go from the graph of y = x² to y = a (x - h)² + k you can use the mapping formula:

(x, y) ⟶ (x - h, ay + k)


Examples:


  1. y = (x - 4)² + 2 (x, y) ⟶ (x + 4, y + 2)
  2. y = (x +7)² (x, y) ⟶ (x - 7, y)
  3. y = 5 (x + 2)² - 3 (x, y) ⟶ (x - 2, 5y - 3)
  4. y = ½ (x - 3)² + 1 (x, y) ⟶ (x + 3, - ½y + 1)
  5. y = - 2x² + 6 (x, y) ⟶ (x, - 2y + 6)

UNIT 1: FACTORED FORM

UNIT SUMMARY


  • Expanding & Simplifying
  • Factored Form
  • Factoring common factors
  • Factoring Common Binomials
  • Factoring by Grouping
  • Special Cases
    - Perfect Squares
    - Difference of Squares
  • Factoring Simple Trinomials
  • Factoring Complex Trinomials
  • Solving word problems

Learning Goals

  • Learn how to expand and simplifly
  • know the factored form
  • learn hot to get factored form
  • learn how to factor binomials
  • know how to factor by grouping
  • learn special cases
  • know how to factor simple trinomials
  • know how to factor complex trinomial
  • learn how to solve word problems

EXPANDING AND SIMPLIFYING

Expanding and simplifying binomials is a process that simplifies factored form. Expanding factored forms will give us our simplified trinomial or binomial. To expand we can use the algebra tiles method or the F.O.I.L method. For the algebra tiles method we just complete the square to get our answer. For F.O.I.L it's a much long process. F.O.I.L stands for First, Outer, Inner and Last. We first multiple the first terms in the factored form then we multiple the outer terms, inner terms and finally the last terms After we're done expanding we simplify by collecting like terms. The video below will help you visually with expanding using algebra tiles and the F.O.I.L method.
Multiplying Two Binomials Using the FOIL Method

FACTORED FORM

The function for factored form is y = a (x - r) (x - s)

The values of r and s are the x-intercepts/ roots/ zeros and are written like: (r, 0) (s, 0)

The value of a will tell you the shape and direction of the parabola. Just like in vertex form, the a value determines the stretch or compression of the parabola (how wide or narrow it is). If the value of a is positive, the parabola will open upwards. If the value of a is negative, the parabola will open downwards (reflect onto the x-axis).

COMMON FACTORING

COMMON FACTORING


Common factoring is the opposite of expanding. Expanding involves multiplying while factoring involves dividing.


Different ways to Factor :



  1. Finding the GCF
  2. Factor by Grouping



Finding the GCF


When using this method, find the greatest common factor of the polynomial's terms. This will include the GCF of its coefficients and the GCF of its variables.


In this example, the GCF of the coefficient was 2 and the GCF of its variables was y^2

Big image

FACTOR COMMON BINOMIALS

Example: 3x (z - 2) + 2y (z - 2)

In order to factor an equation like 3x (z - 2) + 2y (z - 2), you must think of (z - 2) as one factor. Therefore, you would take out (z - 2) and write it outside the bracket, and write what's left on the inside.
Solution: z - 2 (3x + 2y)
Therefore, a binomial can also be considered a common factor.


Example: -8x (z + 4) - 5y (z + 4)
Solution: (z + 4) (-8x - 5y)

FACTOR BY GROUPING

Some polynomials will not have a common factor in all of their terms. They can sometimes be factored by grouping terms that do have a common factor.

For example: df + ef + dg + eg


  1. Split the equation into two halves (be sure to keep the middle sign with the middle term).

    df + ef / + dg + eg


  2. Group the first two terms together and the second two terms together by placing brackets around them. NOTE: always put a plus (+) sign in between the two brackets and if there is a minus (-) sign, put it on the inside of the bracket.

    (df + ef) + (dg + eg)


  3. f is the common factor in the first set of brackets, and g is the common factor in the second set of brackets.

    f (d + e) + g (d + e)


  4. Notice that what is left in the brackets is the same. Therefore, (d + e) is a common factor.

    (d + e) (f + g)

SPECIAL CASES

Perfect squares will usually follow one of the two patterns:

a² + 2ab + b² = (a + b

a² - 2ab + b² = (a - b

To check if the equation is a perfect square, you simply confirm that you can square the first and last term nicely (resulting in a whole number, not a decimal). Next, you must multiply the value of the first term by the value of the last term and multiply by two. If this gives you the middle term, the equation is a perfect square. If this does not give you the middle term, the equation is not a perfect square and you must try a different factoring method. Your answer is written as either: (a + bor (a - b)², depending on if the middle term is negative or positive.

EXPAMLE 1: x² + 12x + 36

= (x + 6) (x + 6)
= (x + 6)²
This solution fits the perfect square pattern since both x² and 36 are perfect squares, and 12x is twice the product of x and 6. Since all signs are positive, the pattern is a² + 2ab + b² = (a + b)².


Difference of squares will usually follow the pattern: a² - b² = (a + b) (a - b)
And will NEVER follow the pattern: a² + b²
This equation shows a formula for factoring a² - b², the difference of two perfect squares. Notice that the factors are identical except that one is addition and the other is subtraction. An easy way to identify if an equation is a difference of squares is to count the number of terms it has. If the equation only has 2 terms, it is difference of squares.


Examples:

  1. x² - 64
    = (x + 8) (x - 8)
    This solution fits the difference of squares pattern since both x² and 64 are perfect squares and the problem is subtraction. Since this is the case, the formula for difference of squares can be used: a² - b² = (a + b) (a - b).

3.10 Special factoring

FACTOR SIMPLE TRINOMIAL

A simple trinomial has a variable of x and a, b, c are constants where a = 1
We use the method of factoring simple trinomials to change an equation in standard form [ax² + bx + c] to factored form [a (x - r) (x - s)] .The value of c is a product of two numbers that also add up to give you the value of b. For example, in the equation x² + 7x + 10, the numbers 5 and 2 multiply to 10 (c) and add to 7 (b).

Another way to write this equation is:
x² + (m + n) x + mn

= (x + m) (x + n)

Using the example above:
x² + 7x + 10
= (x + 5) (x + 2)

Solving Using The Guess & Check Method:


  1. Look for and remove the common factor (if any)
  2. Choose factors for the constant term (c) that sum to the x term (b)
  3. Check by expanding (recommended)


Examples:

  1. n² - 10n - 16
    = (n - 8) (n - 2)

  2. z² + 8z + 7
    = (z + 7) (z + 1)

  3. t² + 11t - 26
    = (t + 13) (t - 2

FACTOR COMPLEX TRINOMIAL

When factoring complex trinomials, we use similar steps as to when factoring simple trinomials.
Factoring Complex Trinomials

SOLVING WORD PROBLEMS

Solving Word Problems Using Factored Form

STANDARD FORM

SUMMARY

  • Standard Form Equation: y = ax² + bx + c
  • a value gives you the shape and direction of openingof the quadratic
  • c value gives you the y-intercept of the quadratic
  • To get x-intercepts, SOLVE using the "Quadratic Formula"
  • MAX or MIN? Complete the square to get vertex form.
  • Discriminant Formula: x=(b^2 - 4ac).
  • In order to find the x-intercepts use the Quadratic Formula.


LEARNING GOALS

1.Quadratic Formula x-intercepts.

2. I will know how to use the Discriminant to find how many x-intercepts a quadratic function has.

3. I learned how to graph a Quadratic Function using the Quadratic Formula.

4. I learned how to Complete The Square to find the Maximum/Minimum value of the quadratic and its' vertex.

5. I will know how to solve a word problem

QUADRATIC FORMULA

The purpose of the quadratic formula is to find the x-intercepts
Big image
Big image

DISCRIMIANT

The discriminat is the square root part of the quadratic formula. The discriminate determines how many x intercept the parabola will have. If the discriminate is greater than zero the parabola has 2 x intercepts. When the discriminate is less than zero there is no x intercepts. The parabola will have one x intercept if the discriminate is zero
Big image
Big image

GRAPHING STANDARD FORM

Graphing standard is super easy. It is just like graphing vertex form and factored form. We find our two x values using the quadratic formula. Solve for the axis of symmetry and optimal value. Then simply graph.


example of y=3x²+6x-2 graph

Big image

COMPLETING THE SQUARE


  • y = ax^2+bx+c

And the vertex form equation looks like:



  • y = a(x-h)^2+k



This part of the website will teach you how to go from standard for to vertex form.


  1. The first step is to place a bracket in which the a becomes isolated
  2. Move the c value outside of the brackets
  3. Divide the b value and square it. This becomes your new c value inside the bracket
  4. Add the opposite of the c value outside of the brackets
  5. Factorize the bracket
  6. Evaluate the outside term (Remember to apply the a value to the inverted c value)
❖ Completing the Square - Solving Quadratic Equations ❖

How to Solve A Completing the Square Word Problem

How to solve a completing the square word problem

Reflection

Overall this unit of Quadratics has taught me so much and has given me so much knowledge so i can take it too the real world and apply it there.

Starting slow in the beginning i have made big progress and I know understand the three main components of these units.

Connection



  • Goes into Factored Form by finding the axis of symmetry and the vertex. Then sub in the vertex into the vertex form equation.
  • By completing the square, you will be able to convert Standard Form into Vertex Form



Factored Form:


  • By solving for the zeroes in Vertex Form, you could convert it into Factored Form aswell
  • Standard Form can be changed into Factored Form by using the 6 different methods of factoring ( Perfect Square and or Trinomial)



Standard Form:


  • Both Factored and Vertex Form can be changed into Standard Form by expanding and simplifying the equation

ASSESSMENT

This assessment has made me understand the greatest feats that a simple object can reach because of the Motion Word Problems. This can show and provide the accurate data to see what a ball, for example will reach and what time it will reach it. This is used in may real world practices
Big image