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
To find the optimal value of this equation
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
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.
Transformations ( Translations Vertical or Horizontal, Vertically stretched, reflection
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
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).
h= 4x^2 + 5t +15
- First step is to determine your a,b and c value. a= 4, b= 5 and c= 15
- Sub in the values into the quadratics formula -4±√ 5^2 - 4(4)(15)/2(8)
- solve -4 ± √10 + 240/ 8
- -4 ± √ 230/8
- -4 ± √15.16/8
- -4 + 15.16/8 = 1.45
- -4 - 15.16/8 = -2.39
Completing the squares
Now we can solve a Quadratic Equation in 5 steps:
- Step 1 Divide all terms by (the coefficient of x2).
- Step 2 Move the number term (c/a) to the right side of the equation.
- 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.
- Now you have something like this (x + p)2 = q, which can be solved rather easily:
- Step 4 Take the square root on both sides of the equation.
- Step 5 Subtract the number that remains on the left side of the equation to find x.
Example: x^2 + 4x - 32
- Step one you get the 32 and you divide it by 2 then you square root it you should get positive 4
- Step two You find a number times 4 to give you 32 which is 8
- Step three The equation you have now should be - 4 + 8 = 4 then (-4) (8) = -32
- 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)
Graphing From Standard form
- Factor the equation.
- Set each bracket equal to 0 to determine the x - intercepts roots and zeroes.
- Find the axis of symmetry and the x - intercepts.
- Sub in the A.O.S into either the standard equation.
- Graph using the x , y and the vertex.
Discriminant is a number inside the square root (b²-4ac) of the quadratic formula. It shows how many solutions a quadratic equation has.
- D>0, 2 x-intercepts
- D<0, No x-intercepts
- D=0, 1 x-intercept