Quadratics

Everything you need to know

Things to be learned in this unit

Intro to Quadratics

Using Tables of Values, Technology and First Differences to Analyze Quadratics

Investigating Vertex Form

Graphing Vertex Form

Finding Equation and Analyzing Vertex Form

Factored Form

Factor Special Trinomials

All Kinds of Factoring + Assessment For Learning

Solving by Factoring

Completing the Square

Solving from Vertex Form

Quadratic Formula + Discriminant

Introduction

What is quadratics?

The word Quadratics comes from "Quad", referring to a square (quadrilateral). This is because in a quadratic equation, the variable gets squared (e.g. X^2)

Important Terms you need to know

In quadratics we use parabola's to find points

Direction of opening: In when graphing a parabola the graph can either be facing "upward" or "downward" this depends of the "a" value in the standard equation which is ax^2 +bx+c. If the value of "a" is negative then the parabola will be facing downward, if the "a" value is positive the parabola will be facing upwards.

Axis of symmetry: Thus is the vertical half point that separates the parabola into 2 equal parts (written as only a number on the x-axis).

Vertex: The vertex is either the highest point on the parabola or the lowest point.

Zeroes: Are points on the graph where the parabola touches or crosses the x-axis.

Optimal Value: This is the "y" coordinate of the vertex

Minimum Value: This is known as the vertex (if the vertex opens upward)

Maximum Value: This is known as the vertex(if the vertex opens downwards)

X-intercepts: The point(s) when the parabola crosses the x-axis

Y-intercepts: The point(s) when the parabola crosses the Y-axis

Finding the tables of values

We learned that finding the first differences from a table of values will tell you whether or not the graph is linear or not (it depends if the first differences are the same). Today, we are going to learn that if there is a second difference that is the same then the graph is quadratic.

How to find first and second differences:

1.When you are given a table you subtract the first value by the second and see what the difference is between the two numbers (you continue this until all the first differences are found).

2. If the first differences are different then you repeat the first step which is to subtract the first number by the second. There are three different type of outcomes, either the first differences are the same (linear), the second differences are the same (quadratic) or none of the differences are the same (no-relation).

Finding first and second differences

Basics to vertex Form

The basic vertex form quadratic equation is written as y = a ( x - h ) ^ 2 + k.


1. In this equation, the a value is the parabola's vertical stretch

- If a is a negative number, the parabola is flipped (opens downwards) or in other words,there is a reflection in the x-axis

2. The h value is the parabola's horizontal translation

- If h = 3, the parabola's axis of symmetry (the middle point) would be at -3 on the x-axis

- If h = -7, the parabola's axis of symmetry would be at 7 on the x-axis


3. The k value is the parabola's vertical translation

- If k = 3, the parabola's optimal value would be at 3 on the y-axis

- If k = -10, the parabola's optimal value would be at -10 on the y-axis

4. By looking at both the h and k value, you can combine both of them to determine the vertex of the parabola

-If the h value is -5 and the k value is 9, the vertex is (5, 9)

Vertex Form

  • Axis of symmetry (x=h)
  • Optimal Value (y=k)
  • Transformations (translation vertical or horizontal, vert stretch, reflection
  • X-intercepts or Zeroes (sub y=0 and solve)
  • Step Pattern
Things to know about vertex form

GRAPHING VERTEX FORM

GRAPHING VERTEX FORM

With the basic equation y = a ( x - h ) ^ 2 + k , we can plot a parabola using just the information from the equation.


1. As mentioned above, the h value, which is always opposite (negative value is positive on the graph and vice versa) tell us the x value of the vertex


2. The k value, which is NOT opposite, tell us the y value of the vertex

3. The a value will tell you the parabola's vertical stretch. If it is negative, the parabola opens downwards. To use the vertical stretch, you must use the step pattern (one left/right and one up/down. two left/right and four up/down) and multiply the number on the up/down. (e.g. A vertical stretch of 2 is one left/right and two up. two left/right and eight up)



  • To go from vertex form to the equation, first find the vertex's x and y value.
  • The x value is the h. Make sure to make it is negative if it's positive and vice versa
  • The y value is the k. This one is not opposite
  • To find the a value, count once right or left of the vertex and count how many times you have to go up or down until you reach a point where the parabola intersects with both the y and x axis lines. This number is your a value. Your a value will be negative if the parabola opens downwards
  • Finally, input the values in the basic equation and you have the answer
How to graph from vertex form

Optimal Value

The optimal value is the highest or lowest point of the parabola. The optimal value is represented by the k in the equation. If the k value is negative then the vertex will be below the x axis and if the k value is positive then it will be above the x axis
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Axis of symmetry

Every parabola has an axis of symmetry which is the line that runs down its 'center'. This line divides the graph into two perfect halves.

The axis of symmetry in factored form uses the formula x= ( (r+s) /2). The x is the axis of symmetry. The axis of symmetry uses the r and the s from factored form. Factored form is y= a (x-r) (x-s).
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Transformations

Transformations are the key to graphing and explaining where the parabola is. It is only used in vertex form because each letter except x and y represents a transformation in this equation y=a(x-h)^2+k.

h = the vertex of the parabola will move to the right or left side of the graph. (Negative numbers move right and positive numbers move left) This is called a horizontal translation right or left depending on the way it goes.

k = the vertex of the parabola will move up or down. (Positive numbers move up and negative numbers move down) This is called a vertical translation up or down depending on the way it goes.

a = how steep the graph is. If a is higher than one the graph will get steeper. This is called a vertical stretch.
If a is less than one but higher than zero the graph will become less steep. This is called a vertical compression.

If a is a negative the parabola will flip. This is called a vertical reflection because the parabola is the same but is just flipped, therefore making it a reflection.

X-intercepts or Zeroes (sub y=0 and solve)

To find the x-intercepts of the equation you would need to set Y=0 so that you can isolate and solve for X.


y=2(x+2)^2-8

0=2(x+2)^2-8 Sub Y for 0

8=2(x+2)^2 Bring 8 over to the other side

8/2=(x+2)^2 Divide both sides by 2

4=(x+2)^2 Square both sides by - and +

+-2=x+2 calculate for the X's

-2=x+2

-2-2=x

x=-4

2=x+2

2-2=x

x=0

The Step Pattern

The step pattern is also a very important thing that you need to know. The step pattern is the movements that you follow from the vertex to find the other points of the parabola so that you can connect them and create a parabola.

Step Pattern:

over 1 up 1

over 2 up 4

over 3 up 9

etc...

Also a key piece of information is that when you are about to use the step pattern you must multiply the up number by your a value so if your a value was 2 then your step pattern would be

over 1 up 1x2

over 2 up 2x2

Mini-Test 3.1

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Factored form

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Factored Form: Finding x-intercepts

The basic equation for the factored form is y =a ( x + n) ( x - n ) , where n is equal to any number. Much like the h value from the vertex form equation, n is always opposite in the factored form.

Finding the x-intercepts:

For this equation you will be finding the zeros (points on the graph where the parabola touches or crosses the x-axis). In this equation the "y' value is always going to be zero because we are looking for our two x-intercepts, eg. (0,5) (0,3)

For example: y=2(x+3)(x-9)

1. You change "y" to 0

0=2(x+3)(x-9)

2. We aren't really concerned about the "a" value as of yet, we just want to find the x-intercepts. Therefore next we have to write (x+3)=0 and (x-9)=0

3. Now you have to do basic solving to find the x-intercepts

x+3=0 x-9=0

x=-3+0 x=9+0

x= -3 x=9

Keep in mind, x-intercepts are written only for the x values

How to find the axis of symmetry (factored form)

To find the axis of symmetry you are going to have to use the formula axis of symmetry=x-intercept + x-intercept divided by 2, this will give you the axis of symmetry.

For example: The x-intercepts we found in the previous equation was x= -3 and x=9

Then you have to pug these values into the formula

A.O.S= -3+9/2

6/2=3

Therefore you A.O.S is going to be three

Basically, we found the number that separates the x-intercepts evenly

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How to find the optimal value (factored form)

To find the optimal value of any standard form equation you should first find the axis of symmetry and then substitute it into the equation to find the last value which is the optimal value. The optimal value is the y coordinate of the vertex. The optimal value represents the maximum/minimum point on the parabola and is needed to finish the equation, or finish graphing it.

The optimal value is the "y" coordinate of the vertex. To find the optimal value all we need to do is substitute the axis of symmetry (refer above) into the equation as the "x's"

For example:

y=2(x+3)(x-9)

y=2(3 +3)(3-9)

y=2(6)(-6)

y=2(-36)

y=-72

Therefore the optimal value is -72

Factoring from standard form to create factored form

There are 6 different types

Common Factoring

Binomial Factoring

Simple trinomials

Complex trinomials

Perfect square trinomials

Differences of squares

Common Factoring

To common factor it is fairly easy, the goal is to change the standard form into factored form (this will be explained later). We are basically breaking down the equation.

For example: (2x+6)

The goal is to put this equation into factored form

1. (2x+2*3) First thing we do is see if the number without the variable can be broken into two numbers. From this when we break the number 6 into two numbers we can see that the 2 is common in the equation

2. After we find this we have to remove the two because it is common.

2(x+3)

3. As you can see we leave whatever is left in the bracket (not common)

Therefore the equation (2x+6) can be factored into

2(x+3)

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Factoring By Grouping

Binomial Factoring

Binomial factoring is when two equations are being multiplied together

In this case you expand and simplify

For example:

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Factoring Simple Trinomials

A simple trinomial is an equation that has three different terms, and the value of a is one, in the standard form equation (y=ax^2+bx+c)


to factor you need two brackets, and both brackets need to equal to the equation.

Easy rule to follow is multiply to C and add to B.

Factoring Simple Trinomials

Factoring Complex Trinomials

Complex trinomial factoring is a process used when you have an equation in standard form but the coefficient at the beginning of the equation in front of the a is greater than one
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How to do Complex Factoring

Perfect Squares

So a Perfect square is when you square root the first and last term, and when you do that you multiply them and then multiply your product by two, which should give you your middle term.
An Example would be: 16m^2+24m+9
So the first terms square root is 4m and the last terms square root is 3. So when you multiply those two you get 12m and multiply that by 2 and you get 24m. So indeed that is a perfect square. Now when you factor it, you should get (4m+3)^2
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How to factor perfect square trinomials

Differences of Squares

Difference of squares is similar to perfect square only instead of the same bracket one of the signs change and the numbers stay the same, because you need one positive and one negative to make the them cancel out

example:

(a+b)(a-b)

a^2-ab+ab-b^2

this equals to a^2-b^2 because ab get cancelled out with each other.

How to do differences of squares
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Solving by factoring

"To solve" in this sense means to find the x-intercepts (basically solving the equation). The most common way to solve an equation is by factoring. As taught above there are many ways to factor (simple factoring, grouping ect.). The only difference now is that after the equation is factored we solve to find the x-intercepts.

For example: x^2-5x+4=0

Step one: First you must analyze what type of equation this is, and what methods are you going to have to use to factoring this equation. In this case since this is a simple trinominal you must use the same steps to factor a simple trinominal.

x^2-5x+4

-4

-1

Therefore the factored form of this equation would be...

(x-4) (x-1)

Step 2: Now after you the equation is factored you must find the x-intercepts. This is fairly simple because the x-intercepts for this factored equation is just the opposite of the number used in the equation.

x=4 x=1

The way how I found this was by solving for x

x-4=0 x-1=0

x=4 x=1

As you can see, I moved the number over to the right side(changed the sign) and got my answer.

Completing the square: from standard to vertex form

To "complete the square" it is fairly easy. The main goal is to go from standard form to vertex form. Standard form being y=ax^2+bx+c to the equation y=a(x-h)^2+k. There are a few steps that is mandatory to take to successfully change standard form into vertex form, and these are the following.

Example: X^2 +18x+81

Step 1. Place two brackets around the two values with x's isolating the c value (x^2+18x)+81

Step 2. Divided the b value by 2 then square root the number you find and place it into the equation (keep number inside the bracket). When you divided the number keep it inside the bracket but have the same number as negative and positive because you can't add any new numbers into the equation and it equals zero.

(x^2 +18x+81-81)+81

Step 3. Find the expression that represents the three numbers found inside the bracket. This can be easily done by dividing the b value by 2.

(x+9)^2

After this is done take out the negative number and place it outside the bracket and add or subtract the number numbers together.

(x+9)^2+81-81

Step 4: Once you are finished,determine your new vertex which in this case would be (-9,0)

That is how you change from standard form to vertex form.

Solving x-intercepts from vertex form

To solve form it is fairly simple once again. There are steps that have to be taken to be able to find the x-intercepts from a vertex form. The main goal from is to find the x-intercepts and see that the value can be negative or positive.

Example: y=2(x-5)^2-50

Step 1: Change the value of "y" to 0 in order to find the x-intercepts

0=2(x-5)^2-50

Step 2: Bring the k value over to the left side of the equation

50=2(x-5)^2

Step 3: You want to get rid of the a value in front of the bracket therefore you divide that number by itself (and whatever you do on the right side you do to the left side) therefore you do the same on the other side

50/2=2/2(x-5)^2

Step 4: After that you want to get rid of the square for the x and h bracket, you do so by square rooting it which would be the same thing without of square sign. Remember what ever you do to the right side you have to do to the left side, therefore you do the same.

{25=(x-5)

Step 5: When you square root any positive number (keep in mind that you can only square root positive numbers) there can be two values either negative multiplied by negative or positive and positive. Therefore you must identify that there are two possible answers.

5+ - = x-5

Step 6: Isolate x by using both values, therefore you will see both outcomes and have your two x-intercepts.

5=x-5 x-5=5

x=10 x=0

That is how you find the x-intercepts from vertex form

Quadratic formula + Discriminant

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The quadratic formula is used for finding the x-intercepts also called roots. The formula works the same as regular formula's because all you have to do is plug in the values and solve. This formula only works for standard form (ax^2+bx+c) equations because the values from standard form (a,b and c) are that fit the formula.

For example:

Say we have the equation 4x^2-11x-3

Step 1: We label our values which are a,b and c

4=a -11=b -3=c

Step 2 :Once we have that we substitute these values into the equation

x=11+-{-11^2-4(4)(-3)/2(4) (keep in mind that I changed the sign because the sign rules apply here. The rule is - x- =+, +x+=+, -x+=-)

Step 3: You solve to find the discriminate (this is the number inside the square root sign. If the discriminant is 0 then there is only one answer, if the discriminant is negative then there is no answer because it isn't possible to square root a negative number and if discriminant is positive then you square root the number and continue to solve for the x's.

x=11=-{121+48/2

x=11- +{169/2

x=11 - +13/2

Step 4: Once you found the discriminant then you continue to solve for the x-intercepts. The reason why there is a plus and minus sign is because the x-intercepts can either be negative or positive. Another reason why there is a plus and minus sign is because when the number inside the square root is square rooted then it can be either positive or negative (-x-=+ and +x+=+). Therefore you have to do both roots one negative and one positive.

x=11-13/2 x=11+13

x=-1 x=3

Word Problems

Word problems are the same thing that you have been doing just worded, hence the name word problems. The trick to solve word problems is you have to be able to unriddle and understand what the word problem is asking you for.

Example #1

The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h = -5t^2 + 20 t+ 60, where t is the time in seconds, and h is the height in meters.


a) Write the above formula in factored form

h=-5t^2 +20t +60

h= -5(t^2 -4 -12)

h= -5(t-6) (t+2)


b) When will the rock hit the water?

h= -5(t-6)(t+2)

t-6=0 t+2=0

t=6 t=-2

The rock will hit the water at 6 seconds because that is the only positive time. You can't have a negative time it is impossible. Therefore the rock hits the water at 6 seconds.

Example #2

The path of a ball is modelled by the equation y= -x^2+4x+1, where x is the horizontal distance, in meters, travelled and y is the height, in meters, of the ball above the ground. What is the maximum height of the ball, and at what horizontal distance does it occur?
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Example #3

The Height of the ball is thrown following the path h= -5(t-3)^2 + 46.5 where h is the height in meters and t is time in seconds

a)What is the maximum height of the ball?

46.5 (This is the maximum height because the h value is considered as the maximum height)

b)How long does it take the ball to reach its maximum height?

It would take the ball 3 seconds (I took this right from the vertex because the k value is considered as the time in word problems)

c)What was the initial height of the ball when it was thrown?

h=-5(t-3)^2+46.5

h=-5(0-3)^2=46.5

h=-5(-3)^2 +46.5

h=-5(9)=46.5

h=-45+46.5

h=1.5

Therefore, the height of the ball when it was thrown was 1.5 meters

Reflection

This was just the basics there is still much to learn, but I'm glad you made it to the end, did any of this help? Let me know!! This unit was actually a fun unit surprisingly, I had a lot of fun learning about different equations and different formulas. I hope this unit benefits me and many others in future studies. I have learned a lot of tips and tricks to be able to solve quadratic equations and I hope next year is as good as this one!

Here's a quiz that you can do to help improve your understanding!

Two simple review's that will help you practice

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Helpful sites to check out!