Quadratic Relationships

By: Dylan Seenarine

What is a parabola ?

A parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape. You will know that it is a quadratic equation if it has a x².

Parabola In The World Today

parabola's in the real world

Overview Of Unit

In the quadratic relationship unit we learned many ways and methods to graph and understand many types of equations.

Components of Parabola


Axis Of Symmetry : The A.O.S. divides the parabola in 2 halves, it is a line of symmetry for a graph. It is when the two sides of a graph on either side of the axis of symmetry look like mirror images of each other. It can be found on the x-axis.
Optimal Value: The lowest or highest point the parabola can go. this is the y value in the vertex. The optimal value determines if the parabola goes upward or downward (max or min).

Vertex: The maximum or minimum point on the graph. The vertex consists of the axis of symmetry, which is the x coordinate and the optimal value, which represents the y value.

Y-intercept: The y-intercept is point on the parabola that hits y-axis

X-intercept: The x-intercept is the point on the parabola that hits the x-axis

Zeros and Roots: these are the x values and is when you set the y=o

VERTEX FORM: Y=A(X-H)²+K

QUADRATIC RELATIONS IN VERTEX FORM: Y=A(X-H)2 +K

(h, k) is the vertex of the parabola, and x = h is the axis of symmetry.
• the h represents a horizontal shift (how far left, or right, the graph has shifted from x = 0).

• the k represents a vertical shift (how far up, or down, the graph has shifted from y = 0).

• notice that the h value is subtracted in this form, and that the k value is added.
If the equation is y = 2(x - 1)2 + 5, the value of h is 1, and k is 5.

If the equation is y = 3(x + 4)2 - 6, the value of h is -4, and k is -6.

EXAMPLE OF VERTEX FORM

Example 1: y= (x+3)²


Vertex: (-3,0)


Axis of symmetry: X=-3


Stretch or compression factor relative to y=x²: None


Direction of opening: Upward


Value of x: X=-3


Value of y: Y=0

Example 2: y=(x+2)²+5

Vertex: (-2,5)


Axis of symmetry: X=-2


Stretch or compression factor relative to y=x²: None


Direction of opening: Upward


Value of x: X= -2


Value of y: Y=5

Example 3: y=3(x+7)²-2

Vertex: (-7,-2)


Axis of symmetry: X=-7


Stretch or compression factor relative to y=x²: 3


Direction of opening: Upward


Value of x: X= -7


Value of y: Y=-2

FINDING THE ZEROS IN VERTEX FORM

Step 1:Find the vertex. Since the equation is in vertex form, the vertex will be at the point (h, k).

Step 2:Find the y-intercept. To find the y-intercept let x = 0 and solve for y.

Step 3:Find the x-intercept(s). To find the x-intercept let y = 0 and solve for x. You can solve for x by using the square root principle or the quadratic formula (if you simplify the problem into the correct form).

Step 4:Graph the parabola using the points found in steps 1 – 3.

Quick Way of Graphing a Quadratic Function in Vertex Form

VERTEX FORM Using STEP PATTERN

The step pattern is a pattern that identifies your next points in each parabola.

The original step pattern r is 'Over 1 up 1 and Over 2 up 4'

But in quadratic relations if there is an 'a' value the step pattern change.

Common Factoring

Common factoring should always be the first thing you look for when you have an equation in standard form. You should look to see if each of the terms in the equation has a similar component that each of the terms can easily be divided into. The common factor can either be a variable (x) or a coefficient (any number.)


Common factoring:

Example 1: 8x + 6

Step #1: Find GCF ( what is the greatest # in common between 8 and 6)

GCF=2

Step #2: write solution with brackets

8x/2 + 6/2 = 4x + 3

= 2(4x + 3)


Example 2: 12x³ - 6x²

step #1: GCF= 6x²

Step #2: 6x²(2x-1)

check your answer!

6x²(2x-1)

=12x³ - 6x²

FACTORED FORM: Y=A(X-R)(X-S)

y= a(x-r)(x-s) is the equation of a factored form.


You have to multiply everything within the brackets by each other before multiplying it by "a".


(x)(x) (x)(s) (r)(x) (r)(s)

After multiplying these you then simplify and then multiply it all by the a value

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Expanding

Expanding is used to expand brackets and to simplify


example:


4(3+4x)

- multiply the number outside the bracket by everything inside the bracket

= 4 x 3 + 4 x 4x

= 12 + 16x

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Factoring Simple Trinomials

What I like to do for this type of factoring is just use mental math but others may find it difficult. To factor a simple trinomial you need to find 2 numbers that multiply to give you the C value but also add up to give you the B value


For example:


x^2 + 7x + 6



6 and 1 are factors of 6 that will equal 6 if multiplied and 7 if summed.


You will then use these 2 numbers part of the equation y=a(x-r)(x-s)


y=(x+6) (x+1)

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Factoring Complex Trinomials

This type of factoring is a bit confusing so I'll break it into steps.


1. You first need to multiply the A value with the C value.

2. Next, find the common factor of the product you get and find 2 numbers that will add together to give you the B value

3. After you find the 2 numbers sub them into the equation and group them.

4. Find common factors of each group and you will then use the numbers inside the brackets as well as the 2 numbers you got from grouping.


Example:


2x2 – 3x – 35


2(-35) = -70

Factors of -70 that add up to give you -3 are -10 +7

Now sub those into the equation


2x2 – 10x +7x – 35


Find the GCF of each group you create


2x(x-5) + 7(x-5)


Now put them together and your answer is:


(2x+7)(x-5)

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Difference of Squares

When working with a difference of squares, you must check to see that
1. Each term is a squared term and
2. There is a minus sign between the two terms.


Two terms that are squared and separated by a subtraction sign like this:

a2 - b2

Useful because it can be factored into (a+b)(a−b)


A difference of squares is always a binomial. Both terms in the binomial have a square root but the second term is always a negative


Example:


9x^2-49


(3x-7) (3x+7) will give you the exact same answer

Factoring the Difference of Two Squares - Ex 1

Quadratic Formula

The quadratic formula is used to factor a quadratic equation. The quadratic formula can be used for any equation, unlike factoring, which is sometimes too complicated to do without the help of a formula, like the quadratic equation!



Using the quadratic formula is simple, all you need to do is input numbers from the standard form equation to the quadratic formula, and then solve the formula

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Perfect Squares

What a perfect square?

Perfect Square Trinomial is the product of two binomials. But, both the binomials are same. When factoring some quadratics which gives identical factors, that quadratics are Perfect Square Trinomial.


Example of a perfect square:

x² + 10x + 25

= (x + 5)(x + 5)


= (x + 5)²

Factoring perfect square trinomials

Factoring by grouping

Shown above in complex trinomial factoring, grouping is done by finding the "Greatest Common Factor" (G.C.F.) of terms

Example:

Factoring by Grouping - MathHelp.com- Algebra Help

Standard form To Vertex form

A standard form equation can be converted to a Vertex form equation, this helps to graph the equation with ease.


Example:


2x^2 + 16x + 32

2(x^2 + 8x) + 16

2(x^2 + 8x + 16) - 32 + 16 (Divide b by two then square it (8/2^2))

2(x^2 + 8x + 16) -16 (Add in the opposite of the Squared number and multiply it by the coefficient in the front of the equation)

2(x + 4)^2 - 16

Word Problems

In Quadratics there are many types of word problems. Here are a few examples :


Example. One number is the square of another. Their sum is 132. Find the numbers.

Let A and B be the numbers. The first sentence says one is the square of the other, so I can write


The sum is 132, so


Plug into and solve for B:


The possible solutions are and .

If , then .

So two pairs work: -12 and 144, and 11 and 121


Example. The difference of two numbers is 2 and their product is 224. Find the numbers.

Let x and y be the numbers. Their difference is 2, so I can write


Their product is 224, so


From , I get . Plug this into and solve for y:


If , then .

If , then .

So two pairs work: -14 and -16, and 14 and 16.

Linking Topics together

Almost all topics that have been learned can be linked to graphing. Vertex form was used to graph parabolas. Even if the equation was standard form you could "complete the square" and change the equation to vertex form and graph it. This is one of the ways the topics can be used with graphing. You can even use factored form to find the x intercepts and then graph them. With the x intercepts found you can then find the vertex.

Reflection On Quadratics

Quadratics was very interesting though very challenging for me to learn and understand. I never really got the concept of parabolas because they were quite difficult for me from the beginning until the the end of the unit. My tests and quizzes were not done to the best of my abilities either instead done very poorly. Quadratics was similar to what we learned in grade nine though a great leap further. In grade nine it was linear lines and now their curved lines. We learned quite a lot this unit. Such as, common factors, multiplying binomials, regular and complex trinomials, completing the square, perfect squares, differences of squares, and we also learned 3 equations. We even learned how to convert one equation to another in different forms. The things we learned in quadratics is extremely useful in school and the real world in many scenarios.