By: Anjali Modi
What is Quadratics?
Determining Quadratic Relations
- Before you have learned that to find if it's a linear relation you find the first differences which was with y values and if they are the same value in that row, that shows it a linear relation.
But now to identify if it is a quadratic relation with second differences:
- You determine the first differences, if not the same then you find the differences of the first differences and if that the same its a quadratic relation.
- It is written like: y=a(x-h)²+k
- The basic form is : y=x²
What each Variable Represents
- a represents the stretch/compression of the parabola and the reflection.
- h represents the x-value of the vertex and the axis of symmetry.
- k represents the y-value of the vertex and the Minimum/Maximum value.
- x represents the possible x value that can be represent in the parabola's. Usually it is all set of real numbers because the parabola gets wider and wider so the x-axis can keep extending so that means all the numbers on the x-axis can be a point.
- y represents the possible y values. If the parabola is opening downward then it would be the y-value of the vertex and less because the parabola doesn't go higher then the vertex so no y-values above the vertex would be in the parabola, only numbers below the vertex would be used because the parabola goes through those. If the parabola is opening up then, it would be the y-value of the vertex and higher because the parabola goes through those and not the numbers under the parabola.
Characteristics of a Parabola
- Axis of Symmetry- Divide the parabola into 2 equal halves.
- Vertex- Its the highest or lowest point on the graph depending on the way of the opening.
- X-intercepts/Zeros- Where the parabola intercepts with the x-axis.
- Minimum/ Maximum Value- The highest/Lowers y-coordinate the parabola hits.
- Y-intercepts- Where the parabola intercepts with the y-axis.
- Optimal value- Value of the y-coordinate of the vertex.
- Direction of opening: It can be upward or downward.
- As I have already stated Stretch/Compression is represented with "a" in the equation y=a(x-h)²+k. A stretch has occurred, if the number is higher than 1 and a compression has occurred if the number is less than 1.
- If the "a" is a negative, that means that their has been a reflection on the x-axis. Bu is its a positive that means no reflection has taken place.
- There is vertical and horizontal translation
- The "k" is the vertical translation. This tells us whether the parabola is moving up or down and up which is the vertical stretch. If the parabola moves up that means its positive and if it moves down that means its negative.
- The "h" is the horizontal translation. This tells us whether the parabola is moving left or right which is the horizontal stretch. Usually when you think of this you would say that when is moves left its negative and when it moves right is positive, but its the opposite. If the parabola is moving right then its negative and if its moving left its positive.
Graphing using Step Pattern
- The basic parabola is y=x²
- For any basic Parabola the Step Pattern is always "over 1 up 1, over 2 up 4".
- A basic parabola always has a stretch of one
- Over 1 up 1, Over 2 up 4.
- Not a basic form is a parabola with a stretch or compression
To find the step pattern of a parabola with a stretch or compression (has an "a"), what you would do is...
- First,put the vertex on the graph
- Second, you multiply the stretch/compression ("a") with the basic step pattern. So you would multiply "a" with 1 and "a" with 4.
For example, Let 4 represent "a"...
4x1=4 this show that its over 1 up 4
4x4=16 this shows that its over 2 up 16
- Third, plot the points collected from the step pattern. *So 5 points should be plotted
*** It doesn't matter if it is a fraction, decimal or negative you follow the same step pattern rule of multiplying a with the step pattern ***
Graph of Equation y=2(x+3)²-3
Determining Axis of Symmetry
- Take the x-coordinate of the vertex which is the "h" in vertex form, and that is what the axis of symmetry because this is what divide the Parabola into 2 equal halves going through the x-axis.
- It is written as x=h
- The Axis of Symmetry would be x=-3 on the x-axis.
Determining the Optimal Value
- Take the y-coordinate of the vertex which is the "k" in vertex form, and that's what the optimal value would be because that is the lowest or highest point of the parabola.
- Its written as y=k
- The optimal value would be y=-3, which in this case would be the lowest point.
- This is written as y=a(x-r)(x-s)
- An example would be y=(x-5)(x-4) or y=2(x-3)(x+8)
Determining the X-intercepts/zeros
- To figure out the x-intercepts in factored form you just have to switch the operation of the "r" and "s" and that will give you both points where the parabola intercept with the x-axis.
- Which would be the same as setting each factor to equal 0, giving you x-r=0 and x-s=0.
Determining the axis of Symmetry
- Add both of the x-intercepts together.
- Then, divide if by 2
Determining the optimal value
- Substitute the axis of symmetry into the x in the factored form.
- And then solve for y.
- When you have a stretch or compression it doesn't change.
- You just have to multiply one more time.
Graphing factored form
- Figure out the optimal value, Axis of Symmetry and x-intercepts
- Then plot the 3 points which you have got from doing the optimal value, Axis of Symmetry and x-intercepts.
- Putting the values of the Axis of Symmetry (AOS) and Optimal value together, will give you the vertex
- This is because the x-value is the AOS, where the center point of the parabola is, and the y-value is the optimal value , where the lowest/highest point of the parabola
- So both of these are apart of the tip of the parabola which, is the vertex.
One easy method to multiply binomials is FOIL.
- F- stands for First
- O- stands for Outer
- I- stands for Inner
- L- stands for Last
- First would be X x X
- Outer would be X x s
- Inner would be r x X
- Last would be r x s
Converting into Factored Form
GCF: Is the largest number which can be divided evenly among a set of numbers
Example: The GCF of 27 and 63 would be 9.
- When we common factor we have to examine the coefficients and variables to figure out the GCF.
How do we do common factoring:
- First, find the GCF of the numbers in the expression
- Then, you have to divide the whole expression by the GCF and the GCF will just come before the brackets.
- The GCF was 4 because it is the highest number which can go into both 4 and 12. So we divided the GCF of 4 with the whole equation. Which would be 4/4=1 and 12/4=3. Then put the 4 in front of all the brackets.
- Since the GCF was 5, I divided the whole expression with 5 giving us 15/5=3 and 10/5=2 and for the variables we would put x because we have to find the highest degree of the variable common to each term, which would be x in this case.
- Since the GCF is 3, I divided it with the expression. The x² is in front of the bracket because its the highest common variable.
- To check what you have to do is expand your answer to make sure you get what you started with
Binomial Common Factoring
To Binomial Common Factor you need to...
- Put the common term in one set of brackets and then but the number factored out together in one set of brackets becoming one term
- So first, since (4a+6b) are repeated twice we can just write it once.
- Then, what left would be the 5a and -2b, so we would put it together in one set of brackets.
- Basically their will be 2 sets of bracket one with the repeated numbers and one with the numbers left. giving us, (4a+6b)(5a-2b).
Factoring by grouping
So, what you do would group the terms which have a common factor
- You would group the first 2 terms together and then the last 2 terms together
- Then, you would find the GCF and factor it out.
- *After, you would do whatever you did in Binomial Common Factoring
- So, put the common term in one bracket and put the 2 numbers you factored out in one bracket as one term.
Factoring Simple Trinomials
- You have to find 2 number which gives the product of "c". (You need 2 numbers which multiply together to give you value of "c").
- The same 2 numbers have to have the sum of "b". (They have to add up to "b").
- You factor them into bx
- Then go the factoring by grouping method.
Another way is:
- Find the terms which multiply to give you "c" and add to give you "b"
- Then, you find the numbers which multiply to give you "x²" and but one term in one sent of bracket and the other one in another (x+___ )(x+____ ) or if the term is 2x², it would be (2x+___ )(x+____ ).
- Then the 2 terms which multiply to give you c and add to give you b you would put one number after each "x".
Factoring Complex Trinomials
- Complex Trinomials are different from simple trinomials because for complex trinomials the "a" has a value other than 1.
- All those letter listed below the equation all equal to 8 when multiplied.
- You have to try putting all these numbers in the ? and figure out which one will add up to 19x. To do this you multiply the outer, then the inner and then add those numbers together.
- The 2 ways that would divide 6 are: 6 and 1 & 2 and 3. You have to try dividing the 1st coefficient in different way because you wont get the answer by diving the coefficient in one way. So, in order to figure out numbers you need to divide the coefficient differenly so you get number which will actually work
- Doing perfects is an easier way to do expanding
- So what you have to do is use the Acronym Sam Doesn't Pull String
►D-Double----P-Product (Double Product)
►S-Square the second term
- You can do perfect square if you expand (a+b)² or (a+b)(a+b), because your multiplying the same things that would be squared.
- The variables and values would be different and be part of larger relation
- If the last number has a square root and that number double gives you the answer of the 2nd term. Also, if the 1st term has a square root, that means is should be a perfect square. For the one above, it is a perfect square which is (x+6)² which is the same as (x+6)(x+6).
Differences of Squares
- If you don't have a perfect square you have something like (a-b)(a+b)
- It would be difference of square because the signs are opposite. in one bracket its plus and the other one is negative.
- Square the first term
- then, square the second term
- You put a negative sign because its difference of square.
- It works if its (a+b)(a-b) or (a-b)(a+b).
- It is written like this y=ax²+bx+c
Completing the Square
There are many steps needed to complete the square.
- To change the expression into Vertex Form. You first off, have to take the first two terms and put them into one set of brackets, and then you would leave the last terms outside the brackets. ***The term which you are taking out is the "c" term and this will become your "k" term, if there is one.
- Then, for the brackets you would factor out the GCF.
- With the "b" value, you would divide it by 2 and then square root it in order to get the number witch you need to add and then subtract in the brackets. Then add and subtract that number you got, in the brackets.
- You would then be taking the subtracted number which is the last term in the bracket and multiply that number by the "a" value, to get the number out of the bracket.
- Then you would add or subtract the values outside together so you get the "k" value.
- Then, you will be able to make a perfect square with the values in the bracket. You would keep the "a" value the same
Using Quadratic formula to find the x-intercepts
- Since you are solving to find the zero's, your standard form should be equaling to 0 when using the quadratics formula.
- Since you know the formula and are given the question, all you have to do is plug in the values.
- Plug "a" value in "a", "b" value in "b" and "c" value in "c".
- The discriminant is the number which is being square rooted in the quadratic formula (b²-4ac).
- This is used to tell how many x-intercepts there are...
- If the discriminant is a negative number, that tells us that there are No X-intercepts, because a negative cant be square rooted.
- If the discriminant is a 0, that tells us that there is only 1 X-intercept, because if you add or subtract a 0 the answer would be the same.
- If the discriminant is a positive number that tells us that there are 2 X-intercepts, this is because when adding and subtracting it you would get 2 different solutions.
Axis of Symmetry
- The formula for finding the axis of symmetry is...
x= - b
- All you have to do to find the AOS is sub in the values in the formula
- For the parabola above the AOS is x=0.83, to figure it out i just substituted the values and that how easy this is.
- To find the optimal value which is the y-value and y-intercept in the vertex all you have to do is sub the AOS into the original equation
- Sub the AOS into the x
- I just substituted the AOS in order to get the Optimal value (solve for y)
- In the example above the optimal value is y=-1.2
Graphing in Standard Form
In the quadratics unit I have learned many new things, stuff that were easy and things that were a little more complicated to understand. I learned that in this unit there are things that appear in real life, for example like parabolas when you would look at a rollercoaster you would never think there is any math involved but now when looking at it you will know that they are parabolas and they were built with the math skills. First, one thing that I found difficulty to understand would be perfect squares and differences of square, because when doing it, I could never tell if it is either of these. I would be confused to which one is being used of if it could even be done. Another thing I had difficulty was creating my own question for factoring by grouping. This was complicated for me because I didn’t know how to make them, I thought that you could put any numbers but then I realized that you have to use only specific numbers so this method could work. Another struggle I had was with complex trinomials. At first, I found I hard to find all the possibilities that the value of “a” could be. After I got the hand off it, it became easier because I was getting more comfortable with it. The topics which I understood would be simple trinomials and all the other expanding. The one thing I understood clearly is finding the x-intercepts, AOS, Max/Min value, graphing and word problems that were in Vertex Form. Which is evident on the test below.
I have learned many things about quadratic and I know I will be able to find where this is being used and how it helped on designs become more easier to be made.