Quadratics
By: Dilveer Dhaliwal
How quadratics help us in reality
How to determine Quadratic Realtions
- Recall that linear relations had first differences
So...
- Quadratic relations have SECOND differences
VERTEX FORM
- The vertex form is one way that a quadratic relationship can be written as it is in the form y=a(x-h)² +k and the basic form of it is y=x².
- An example for this is y= 2(x-3)² +5
- The "x" and "y" do not have a value because they are variables for the x and y intercept(s) which will have different values according to the relation
What each variable represents in the Vertex Form
- In the vertex form y=a(x-h)² +k each variable has a part to do
- The a tells us if there is a stretch/expansion or compression to the parabola.
- The h tells us the x value of the vertex and the axis of symmetry.
- The k tells us the y value of the vertex and the optimal value.
Transformations
- The transformations that can occur to a parabola are vertical or horizontal translations, vertical stretches and reflections.
- As we know the a represents vertical stretch and compression if the value is more than 1 than it is a compression and if it is less than 1 it will be a stretch/expansion
- If a has a negative value then there will be a reflection on the x-axis (the parabola will be flipped; downwards)
The h tells if the parabola moves left or right (horizontal stretch). If the h is negative then it goes towards the right and if it is positive it goes towards the left. Basically it will move the opposite of its sign.
The k informs us if the parabola moves up or down (vertical stretch) so if it is negative the graph will move downwards and if it is positive it will move upwards.
Graphing from Vertex Form using the Step Pattern
- y=x² this is the basic parabola.
- The original step pattern is "over 1 up 1 , over 2 up 4" this pattern can be used for any parabola as long as it is the basic (original) one, so it will not have an a value to determine the compression or stretch.
- The basic parabola will always open upwarads
Graphing from Vertex Form when not in Basic Form
- If the vertex form has the "a" in front of it, it is not a basic parabola anymore, it will either have a stretch or compression and will open either up or down
- The value of a will decide if the step pattern will change
An example...
- First, you will graph the vertex
- Then multiply the original step pattern by the value that a had been given
- Then plot the points (vertex, 4 step pattern points).
- It does not matter if there is a decimal, fraction, or a negative number for a, you still have to multiply that number with the original step pattern.
How to find the Axis of Symmetry (AOS)
- The way it is written is x=h, so it would be the h in the vertex form
- The axis of symmetry is the middle point of the zeros; the x intercept of the vertex
- So the AOS of the parabola above will be -2 because it is the midpoint of the parabola and it is the x-intercept of the vertex
How to find the Optimal Value
- It is written as y=k, so it would be the k in the vertex form
- The optimal value is either the highest point or the lowest point on the parabola; the y-intercept of the vertex
- So the optimal value of the parabola above would be -6 because in this case it is the lowest point on the parabola and it is the y-intercept of the vertex
X-intercepts and Zeros
- To find the x-intercept/zeros in vertex form you would have to set y=0
- For instance let's say you have to find the zeros in the equation y=-3(x+2)²+27 the y would be replaced with 0
- We do negative and positive square root because the number being squared could be either one
- As a result the x-intercepts would be 1 and -5
FACTORED FORM
- The factored form is another way that a quadratic relation can be written as and the form is y=(x-r)(x-s)
- An example of this form is y=0.5(x+3)(x-9)
Zero and X-intercepts
- Zeros can be found by setting each factor equal to 0 meaning x-r=0 and s-r=0
- For example in the relationship y=0.5(x+3)(x-9) the zeros would be found by taking each factor (x+3) and (x-9) and make them equal to zero
- Therefore, the zeros/x-intercepts are -3 and 9 for this relation so when you graph the zeros these are the values you would plot for the x-intercepts.
Axis of Symmetry (AOS)
- To find the axis of symmetry you have to add the two x-intercepts and divide it by 2
- Therefore the formula or method you would use is x= (r+s)/2
- For the example we have been using previously we would find the x-intercepts (-3 and 9) add them then divide the answer by 2
- So it would be
x=(r+s)/2
x=(-3+9)/2
x=6/2
x=3
- For that reason, the axis of symmetry would be 3
Optimal Value
- In order to find the optimal value you would have to substitute the axis of symmetry into the equation as x since it is a x value on the parabola
- If we use the example we have been using so far we would substitute x=3 into the equation y=0.5(x+3)(x-9)
- Therefore the optimal value for the example is -18
Graphing
- Now that you know how to find the x-intercepts, axis of symmetry and optimal value you can plot 3 points
- The first x-intercept which is x=-3
- The second x-intercept which is x=9
- The vertex which is (axis of symmetry;x, optimal value;y) (3,-18)
STANDARD FORM
- Last but not least, the standard form is also a way to write a quadratic relation and the form is y=ax²+bx+c
- An example of this form is y=3x²+15x+18
Quadratic Formula
- The quadratic formula is
How to find the zeros using the Quadratic Formula
- In order to use the formula the standard form should equal to 0 since you are solving for the zeros
- Since we know the formula and have the question all you have to do is plug in the appropriate values just like in the example below
- The square root is both positive and negative since there would be a negative and positive square root to the number which in the end would give us two zeros
- If you want to keep the exact answer leave it the way it is in step 4 but if you want to take it down to a fraction/decimal then you can continue and add the first time and subtract the next when solving
Discriminants
- The discriminant is the number that is being square rooted (b²-4ac) this number tells us if there are any zeros/solutions and if there are, how many there are.
- If the discriminant is a negative there will be no zeros because a negative number cannot be square rooted
- If the discriminant is 0 then there is only one zero because it does not matter if you add or subtract a 0 the answer will be the same
- If the discriminant is greater than 0 then it will have two zeros, just as in the example above, since the discriminant is 9 there were two zeros
Axis of Symmetry
- The formula for this is
- To find the axis of symmetry all you have to do is fill in the formula that is shown above
- And do it just as shown in the example below
- It is as simple as that so the axis of symmetry would be x=0.7 for this parabola
Optimal Value
- To find the optimal value (y-value;y-intercept in vertex) all you have to do is substitute the axis of symmetry into the original equation to get y just as we did in factored form
- An example for this is...using the axis of symmetry from above
- So they optimal value for the example we have been using would be -0.45
Word Problem
Completing the Square
- Completing the square helps us turn from Standard form to Vertex form (y=ax²+bx+c to y=a(x-h)²+k
- There are a few steps and they are as follows:
- Take the c out and make it k since they both carry the same value and represent the same thing on the graph
- Then put brackets around the remaining two numbers
- Then factor out the GCF from the number in the brackets; after divide b by two and square to get the number you need to add and subtract
- Then add and subtract that number from the brackets
- When taking the subtraction number out of the brackets multiply it by the a value;then you will be able to make a perfect square
- Make the perfect square (write x and divide b by 2 then put squared outside of the bracket) and write the vertex form
Just as in the example below
- Therefore the vertex form would be y=2(x+2)²-3
Factoring to turn to Factored Form
Factoring-4 terms by Grouping
- If you ever end up with four terms to factor it is very simple
- We group the first two and second two terms
- Then we factor out the GCF from each factor
- Then we can common factor just as shown above
- So basically we grouped the terms into binomials first two and second two, then we found the GCF and factored it out and finally we common factored
- This gave us (x²+3)(x+7)
Factoring Simple Trinomials
- The way that simple trinomials are written is x²+bx+c
- An example for this is w²+5w+6
- There are six steps to get into the factored form and they are
- Make sure the format is x²+bx+c
- Find two numbers that give the product of 'c'
- Add the two number to see if they give the sum of 'b'
- Factor them in for 'bx'
- Do four terms by grouping method
- Then we do common binomials
An example for this is
- Therefore w²+5w+6 in factored form is (w+2)(w+3)
Factoring Complex Trinomials
- Complex trinomials will include an a value with the a making it complex
- An example of this is 8x²+6x-5
- In order to factor this we do the following:
Perfect Squares
- Perfect Squares make expanding easier
- There are 3 steps to do them:
- Square the first term
- Double Product
- Square second term
- But you can only do perfect squares if what you are expanding is either for example (a+b)² or (a+b)(a+b) since you are multiplying the same things that would end up being squared, or (a-b)(a-b) and (a-b)² but of course the variables and values would be different and they can be a part of a larger relation.
Factoring Perfect Square
- You may also use the acronym
Sam
Doesn`t Pull
Strings
which stands for
Square, Double Product and Square
Difference of Squares
- Let's say you don't have a perfect square but something like it for example (y-7(y+7) the only different thing are the signs and this method is called difference of squares
- When we do difference of squares there are only two steps
- Square the first term
- Square the second term
- For example
- The subtraction sign will always stay as subtraction, hence the name difference of squares
- This method can be applied anywhere as long as the relation you are expanding is in the format (a+b)(a-b) or (a-b)(a+b)
Factoring Differnece of Squares
Difference of Squares factors into a product of sum and difference
Common Factoring
- Common Factors are when we factor out the GCF in an expression
- There are only three steps to this
- Find GCF
- Divide expression by GCF
- The GCF number will go outside the brackets as well
Example
2x+4y
2(x+2y)
Since the GCF was 2 we divided the whole expression by 2 and then put the GCF outside of the brackets as well which factored into 2(x+2y)
OR you can make a list on all the factors of each number and find the common one(s) and find the GCF that way; as shown in the example below
Binomial Factors
- This method is known as binomial factors because we are putting the expression into only two parts; an easier way to expand
- For example
- Basically we did three things to get the answer we got
- Since (2s+5t) are the same we can write them just once
- Then we have 3s and -7t left which can be put in another set of brackets
- Both are multiplied so when you need to expand to get back to what you started you may do that
Hence we get (2s+5t)(3s-7t) as our factored form