Quadratics Relationships

Tanish Bhandari

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

Axis of Symmetry (x=h)

The axis of symmetry is a vertical line that intersects the x- axis and divides the parabola into 2 equal parts. It is written as: x=h

Optimal Value (y=k)

The optimal value is a horizontal line that intersects the y axis at the maximum or minimum value of the parabola. It is written as: y=k

Transformations (translation vertical or horizontal, vertical stretch, reflection)

The Vertex form: y=a(x-h)^2+k is used to tell if a parabola has a horizontal or vertical shift, and a compression or stretch.

  • The "h" value tells you if the parabola has a horizontal shift
  • The "k" value tells you if the parabola has a vertical shift
  • If the "a" value is between -1 and +1 and is a decimal or fraction then the parabola has a compression
  • If the "a" value is a whole number more than -1 and +1 then the parabola has a stretch

X-intercepts or Zeros (sub y=0 and solve)

Step Pattern

The step pattern is a pattern that is used to graph, once you find the vertex. Mostly used for vertex form.

When you find the vertex; (h,k) then you can plot that and take the a value in vertex form and multiply it by the step pattern to graph your equation.

Factored Form: a(x-r)(x-s)

Zeros or X- intercepts (r and s)

X- Intercepts (r and s)

A parabola has two x-intercepts, which are described by using r and s. These two variables give you the axis of symmetry by using the following formula; (r+s)/ 2. By using the r and s value you can graph an equation. These are also known as Zeros or Roots.
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Axis of Symmetry (h value: x= (r+s)/2)

The Axis of Symmetry is where the parabola is divided into two equal parts. This is resembled from the following equation; (r+s)/ 2= h. The axis of symmetry is the h value in the vertex. (h,k) ----> (x,y).
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Optimal Value (sub in)

Optimal value is the horizontal axis that intersects the parabola. It is the k value in vertex form. To find the optimal value you sub in the h which is the axis of symmetry into the factored form equation. The y is the optimal value and the y value in the vertex.
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Standard Form: Y=ax^2+bx+c

Zeros (Quadratic formula)

The Quadratic Formula is showed on the right. It is the formula which is used to find the x-intercepts from a standard form. You use the standard form in the quadratic formula to find the zeros.

Example below

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Axis of Symmetry (-b/2a)

This way of finding the axis of symmetry is easier than going through the whole process of the quadratic formula. You can just take your standard form equation, and sub in the following numbers into this equation, and it will give you, your x value of your vertex.

An example on the right is shown for the equation: y=x^2+6x+5

a= 1

b= 6

c= 5

Optimal Value (Sub in)

Once the x value is found, then you can sub that into your standard form equation and solve for y. Example on the right for the equation: y= x^2+6x+5

Axis of Symmetry: -3

Optimal Value: -4

Types of Factoring

Common Factoring

Common factoring is the easiest way to factor equations. You use this method when there is something common in the equation, for example; 2x+20x+40

To factor this i would use the following steps:


  1. Find the common variables or constants
  2. Then factor out those
  3. solve till you can factor fully


2x+20x+40

= 2(x+10x+20)

Since i cannot factor more my answer would be: 2(x+10x+20)

Underneath is an video which explain factoring

3.7 Common Factoring

Simple Trinomials

Factoring simple trinomials is another way of factoring equations. It is used when there is no constant or variable that is common. An example is: x^2+6x+5

To factor this you need to follow the rules below:

  1. multiply the a value by c
  2. find factors of that answer that equal the answer when multiplied, and when added equal the b value
  3. lastly, write them as: (x+/-_)(x+/-_)

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3.8 Factoring Simple Trinomials

Complex Trinomials

To factor a complex trinomial you have to many things. An example is; 12x^2-11x+2

  1. Take the a value (12) and multiply it by the c value (2)
  2. Then find factors of that product (24), something that equals -11 when added. (-3,-8)
  3. Then sub those two values in the b values position. 12x^2-3x-8x+2
  4. Now uses common factoring to factor the equation. 3x(4x-1)+2(4x-1)
  5. Now take the numbers and put them into factored form, and that will be your answer. (3x+2) (4x-1)
Below this is a picture that shows how this was done.

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Completing Perfect Squares

Perfect squares are basically equations being changed from standard to vertex forms.

To complete y= 2x^2+12x-3 use the following steps

  1. Remove the common factor from the x^2 and x term
  2. Find the constant that must be added or subtracted to create a perfect square
  3. Group the three terms that form the perfect square. (move the subtracted value outside the bracket by multiplying it by the common factor first).
  4. Factor the perfect square and collect like terms.
Below is a picture that visually shows how this was done.

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Differences of Squares

Difference of squares is the opposite of perfect squares. It is used when the equation has a subtraction sign in it, for example a Difference of a square is: x^2-49y^2, so a Perfect square would be 2x^2+12x-3

To factor this you find the square roots 1 and 49, then you sub it in: (x+_) (x-_)

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Quadratic Formula

To solve an equation using the Quadratic formula, make sure the equation is in standard form first. Then use the following formula listed below in the picture to solve. Follow these steps to solve this: Y= 2x^2+3x-5


  1. Find your variables: a= 2, b= 3, c= -5
  2. Sub the variables into the Quadratic Formula
  3. Then solve for your x-intercepts
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Discriminants

The number inside the square root (b^2-4ac) Of the quadratic

formula is called the discriminant (D) It help tell us how many solutions a

quadratic equation has.

If D<0 = No Solution

If D>0 = 2 Solutions

If D=0 1 Solution

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