The strategies for adding and subtracting rational numbers can be used to add and subtarct rational expressions : write the expressions with common denominator.
For the example :
So the non-permissible value is x = 0
A common denominator is
This week we learned about equivalent forms.
So In short this chapter.
We learned 3 form to write quadratic function :
First one : General form :
Second one : Standard form :
And the last one : Factored form :
Cause factored form is new one we just learned so i will introduce little bit about it.
Let’s start with example :
And beside how can we change from the general from to factored or standard from. I will show you right now.
This week we started with new lesson : Graphing Quadraction Functions about the determine the value of vertex, formula, what the graph looks like and how to draw the parabola.
So let’s start the things I studies in this week :
: It is general form of quadratic
With a (coefficent) it will helps us know the graph will be big or small.
Also, we can know : If quadratic positive , the parabola will be go up (+) and contrast if quadratic negative , the parabola will go down.
Vertex : highest and lowest point (-1,4)
Axis of Symmetry : which the parabola is symmetric (-1) of above picture
x-intercepts : zero of function or we can determine it by Quadractic Formula
y-intercepts : it depends on c
Maximum point : when the graph opens down Because the intersection point between x and y is at the top
Minimum point : when the graph opens up so the intersection point between x and y is at the bottom
Pattern of Parent Function :
it will show stretch / compress : 1,3,5,7,9..
But if the stretch / compress : 2,4,6,10..
Domain : the value of x and the complete set of possible values of the independent variable, make sure it is real number.
Range : the value of y
: depends on the value of p, It will move to right or left
Let’s star with example :
: when p is positive the vertex move to left
: when p is negative the verter move to right
Today, we will start with perfect square trinomials and The quadratic Formula
First : Perfect Square Trinomials
I will help you know more about that with first example :
We have already discussed perfect square trinomials:
We know : : Square of first term of binomial
2ab : twice the product of binomial’s first and last terms
: Square of last term of binomial
Let’s start with example :
Like my way, i always try the last term the numbers multiply together to get it like this :
I will choose 6×6 because if they multiply together, i will get 36 and when they add together, i will get 12 like the exercise i gave above.
You can do it faster than my way with perfect square.
Answer: (x+6)(x+6) or
Example 2 :
The leading coefficient is not 1 (). Its 9 but Both and 1 are perfect squares, and 6x is twice the product of 3x and 1.
So we will know a = 3a and b = 1.
Then get the answer : or (3x-1)(3x-1)
Quadratic Formula : We will use this formula when we can not use Factoring and Binomial with complicated exercise. This formula will help us get answer easier and faster.
And now let’s start :
We knew : a=2
We will apply this formula :
This week we start to learn the new unit. That’s factoring polynomial expressions
Let’s star with the first example :
If the leading coefficient is 1 like this. The process to do this exercise is so easy. The only two numbers have sum is −4 and that multiply them to give −12 are −6 and 2. So if you wanna find the number have sum = -4 from -12 you can try by this way :
So right now we will chosse one into them. Which ones will help us get -4. That’s -6 and 2 ( -6+2=4 ).
So let’s try with another example :
It has a leading coefficient of 1, find two numbers with a product of 24 and a sum of −10.
Same as above exercise we will try each number multiply to get 24
Which one will help us get 10. Thats last. 4+6
Then we will get . So we got it!!!!
Example 3 :
This example diffirent with two example exercise cause the leading coefficient is not 1 () . We still find two numbers, and those numbers will still add up to 9.
RIght? Cause 10x-1=9x. Its same but i change it to be easy to a simpler equation
Then we will distribute them together then get :
Between the same point is 2x then we will divide each number for 2x to get (x+5)
Same way with -1(x+5)
Finally we will make it simpler by the way, they have same (x+5). We will make the general figures between them.