All About Quadratics

Summary of the Quadratics Unit By Alexia Brown

Table of Contents

1. Vertex Form

  • Axis of Symmetry
  • Optimal Value
  • Transformations(Translations Vertical or Horizontal, Vert stretch, Reflection)
  • X - Intercepts or Zeros ( sub y = 0 Solve)
  • Step pattern


2. Factored Form

  • Zeroes and X - Intercept ( r and s)
  • Axis of symmetry
  • Optimal Value ( sub in)


3. Standard Form

  • Zeroes (Quadratics formula)
  • Completing the squares
  • Simple Trinomials
  • Complex Trinomials
  • Perfect Squares
  • Difference of Squares

Vertex Form

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Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two congruent halves. The axis of symmetry always passes through the vertex of the parabola. The x-coordinate of the vertex is the equation of the axis of symmetry of the parabola.

Optimal Value

The optimal value of a parabola is the minimum or maximum also known as the vertex.


To find the optimal value of this equation

y=0.5x^2+2x+3


To find the optimal value, we need to find the vertex.


In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula x= -b/2a


So the x-coordinate of the vertex is x=-2. Note: this means that the axis of symmetry is also x=-2.


So the y-coordinate of the vertex is y=1.


So the vertex is .


Since the max/min occurs at the vertex as the 'y' value, this means that the min is y=1. So the optimal value is is y=1.

Video on How to Graph parabolas

Step Pattern

Starting from the vertex as "the first point" ...

OVER 1 (right or left) from the vertex point, UP 1² = 1 from the vertex point
OVER 2 (right or left) from the vertex point, UP 2² = 4 from the vertex point
OVER 3 (right or left) from the vertex point, UP 3² = 9 from the vertex point
OVER 4 (right or left) from the vertex point, UP 4² = 16 from the vertex point
and so on ...

where the "Up" numbers are the sequence of "Perfect Square" numbers ...

but always counting from the vertex each time.

X - Intercepts or zeroes

To find the axis of symmetry you must look at the h value of the equation and whats in the bracket is the opposite while graphing .

Transformations ( Translations Vertical or Horizontal, Vertically stretched, reflection

Factored Form

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Optimal Value

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

The optimal value is what determines the maximum and minimum in factored form

Factored form to Standard form

To convert the equation of a quadratic relation from

factored form to standard form, we will expand,

regroup, then simplify the factored form.

Factored form to Vertex form

You do not need to convert this to vertex form from the factored form.

We know that the x-coordinate of the vertex occurs at the average of the 2 zeroes of the Quadratic Function. This function has zeroes at x = -3 and x = 1. So the x-coordinate of the vertex is (-3 + 1)/2 = (-2)/2 = -1.

To get the y-value of the vertex, sub in x = -1 into the original equation to get:

y = -2(x + 3)(x - 1) = -2(-1 + 3)(-1 - 1) = -2(2)(-2) = 8

Therefore, y = -2(x + 3)(x - 1) has it's vertex at (-1, 8).

Standard Form

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Quadratic formula

Formula: -b ± √ b^2 - 4ac/2 square root sign over the b to the c.


h= 4x^2 + 5t +15


  1. First step is to determine your a,b and c value. a= 4, b= 5 and c= 15
  2. Sub in the values into the quadratics formula -4±√ 5^2 - 4(4)(15)/2(8)
  3. solve -4 ± √10 + 240/ 8
  4. -4 ± √ 230/8
  5. -4 ± √15.16/8
  6. -4 + 15.16/8 = 1.45
  7. -4 - 15.16/8 = -2.39

Completing the squares

Steps


Now we can solve a Quadratic Equation in 5 steps:


  1. Step 1 Divide all terms by (the coefficient of x2).
  2. Step 2 Move the number term (c/a) to the right side of the equation.
  3. Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.
  4. Now you have something like this (x + p)2 = q, which can be solved rather easily:
  5. Step 4 Take the square root on both sides of the equation.
  6. Step 5 Subtract the number that remains on the left side of the equation to find x.

Simple trinomials

Factoring Simple Trinomials


Example: x^2 + 4x - 32



  1. Step one you get the 32 and you divide it by 2 then you square root it you should get positive 4
  2. Step two You find a number times 4 to give you 32 which is 8
  3. Step three The equation you have now should be - 4 + 8 = 4 then (-4) (8) = -32
  4. The last equation should be (x - 4)(x + 8) you got the x because you expanded it

Difference of squares

The factors of (a^2 - b^2)


(a + b) and (a - b)


Factor: x^2 - 9

Both x^2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares.


What times itself will give x^2 ? The answer is x.

What times itself will give 9 ? The answer is 3.

These answers could also be negative values, but positive values will make our work easier.


The factors are (x + 3) and (x - 3).

Answer: (x + 3) (x - 3) or (x - 3) (x + 3)

Perfect Squares

A number made by squaring a whole number.


16 is a perfect square because 42 = 16


25 is also a perfect square because 52 = 25

Graphing From Standard form

  1. Factor the equation.
  2. Set each bracket equal to 0 to determine the x - intercepts roots and zeroes.
  3. Find the axis of symmetry and the x - intercepts.
  4. Sub in the A.O.S into either the standard equation.
  5. Graph using the x , y and the vertex.

Dicriminant

Discriminant is a number inside the square root (b²-4ac) of the quadratic formula. It shows how many solutions a quadratic equation has.


Discriminant: b²-4ac

  • D>0, 2 x-intercepts
  • D<0, No x-intercepts
  • D=0, 1 x-intercept