What Is a Quadratic Relation?
A quadratic formula is also sometimes called a second degree formula.
There are THREE quadratic equations:
- Vertex Form
- Factored Form
- Standard Form
How does it apply to everyday life?
An Example of a Parabola
Unit #1: Vertex Form
Summary of Graphing Vertex Form
How to Solve Vertex FormThe equation for the vertex form of a parabola is y=a(x+h)^2+k. If you want to solve with this form you need to substitute y with 0 and solve the equation for x to get the standard form of the equation. If you need to find the y intercept of the equation, substitute the x for 0 and solve to find y.
The Placement of the Parabola
The k and the h in the vertex form determine where the parabola will be on the chart. The k represents the vertical translation of the parabola and the h represents the horizontal translation of the parabola.If the k is positive the vertex of the parabola will be on the upper part of the graph and if it the k is negative, the vertex will appear on the bottom half of the graph. If the h in the formula is positive that means the parabola will be on the left side of the graph, however if it is negative it will be placed on the right side of the graph.
The a determines which way the parabola opens. If the parabola opens downwards, that means that the a is negative. However, if the a is positive, that means the parabola will open upwards
The step pattern is a good way to find out how to graph a parabola. To do the step pattern you must:
· Start at the vertex and go to the next point on the right.
· Identify the rise and the run from the vertex to that point.
· Write them down.
· Then identify the rise and run to the third point.
· Write them down.
· Keep doing so, until you find a pattern in the run. Then do it on the other side to get the parabola.
Axis of Symmetry=(x=h)
Optimal Value (y=k)
Lets Solve a Word Problem
Unit #2: Factored Form: Y= A(x-r)(x-s)
You should also try to be able to make a parabola on a graph using factored form.
Summary of Factored Form
The R and S of the equation are actually the two X intercepts of the parabola.
The A of the equation determines whether the parabola opens up or down and if it is stretched or compressed. If it opens upwards, the A must be positive. If it opens downwards, the A will have a negative value. If the A has a value that is greater than 1 or less than -1, the parabola will be stretched vertically. However, if the A has a value that is in between 1 and -1, the parabola will be vertically compressed.
Different ways of Factoring
There are 5 different types of factoring, they are:
Greatest Common Factor: Monomial Factoring
When we do monomial factoring we factor out the greatest common factor
Simple Factoring: Simple Trinomial Factoring
To factor a simple trinomial we need 2 numbers that multiply to get the last number, but add up to get the middle number as shown below.
Complex Factoring: Complex Trinomial Factoring
To solve complex factoring you need to multiply the lead coefficient with the last coefficient. Once you get that number, you must find two numbers that has a product of that number and the sum of the middle number in the trinomial. Once you get those two numbers, replace the middle term with them. After that, just do common factoring.
2x2 – 3x – 35
2x2 -10x +7x -35
2x(x-5) + 7(x-5)
Difference of Squaresx2 -4
(x-2)^2 or (x-2)(x-2)
(x+6)^2 or (x+6)(x+6)
Here is an in depth video about factoring trinomials
Axis of Symmetry/Optimal Value
There is an equation to find the axis of symmetry using factored form. To get the axis of symmetry you must add up the R and S values and divide them by 2. The equation looks like this: x=r+s/2. Once you solve this, sub this value into the X to get the optimal value or the vertex.
Finding the y intercept
Here is a word problem using factoring!!!
Unit#3: Standard Form: Y=ax² +bx+c
What do the variables mean?
The Quadratic Formula
When do you use the quadratic formula?
Completing the square to get Vertex Form
Here are some word problems for standard form!!!
a) What is the maximum height of the soccer ball?
b) At what time does the soccer ball reach its maximum height?
c) How tall is the building?
When learning vertex form, I learned the characteristics of a parabola like the vertex and y and x intercepts. During the vertex form unit I also learned how create a parabola, name parabola transformations and solve word problems based on quadratics.During the factored form unit, I found out how to expand and simplify quadratic expressions. I also learned different ways of factoring such as complex trinomial factoring, difference of squares, binomial common factoring, perfect square trinomial and factoring by grouping as well as learning to solve factoring word problems. The standard form unit taught me how to find the zeros from the discriminant and how to use the quadratic formula.
This unit has really strengthened my knowledge of graphing, lines and parabolas. Throughout this unit my ability to do word problems has been strengthened and I find graphing much easier. I now also can relate parabolas to real life. I really enjoyed the quadratics unit because of all the new concepts that I have learned.