# Quadratics

## Parabolas

- parabolas can open up or down
- the zero of a parabola is where the graph crosses the x-axis
- "zeros" can also be called "x-intercepts" or "roots"
- the axis of symmetry divides the parabola into two equal halves
- the vertex of a parabola is the point where the axis of symmetry and the parabola meets. It is the point where the parabola is at its maximum or minimum value
- minimum = parabola opens up
- maximum = parabola opens down
- the optimal value is the value of the y co-ordinate of the vertex
- the y-intercept of a parabola is where the graph crosses the y-axis

Understand the base graph, what makes the graph curve, and the step pattern.

## Base graph

## Vertex Form

**y=a(x-h)**2

**+k**

## What Effects Does Each Part of the Vertex Equation Have On The Graph?

**2**

__Vertex equation:__y=a(x-h)**+k**

a = stretch or compression

- stretch is a whole number
- compression is a decimal/fraction that is less than one
- it either opens up (positive) or down (negative)

h = x value of vertex

- moves left and right

k = y value of vertex

- moves up and down
- optimal value

for example:** y=(x-4)**2**+6**

If you are given the vertex points and another point, you can figure out what the 'a' value is

## Try this out!

__State each term for the parabola on the right:__

Vertex -

Direction of opening -

Max or min value -

Axis of symmetry -

Quadratic relations equation in vertex form -

**Ready for the answer?**

Vertex - (2,1)

Direction of opening - up

Max or min value - minimum, y=1

Axis of symmetry - x=2

Quadratic relations equation in vertex form - y=(x-2)2+1

## Try this out!

__Write an equation of a parabola that has a vertex at (2,1) and goes through the point__

__ (3, -7).__

**Ready for the answer?**

y=a(x-h)2+k

y=a(x-2)2+1

-7=a(3-2)2+1

-7=a(1)2+1

-7=1a+1

-8=1a

-8=a

Therefore, the equation is: y=-8(x-2)2+1

## Try this out!

__A football kicked into the air follows a parabolic path described by the equation:__

**h=-2(t-3)2+19 ***h* is verticle height in meters, and *t* is time in the air in seconds.

a) when does the football reach its maximum height?

b) what is the maximum height reached?

c) what is the height of the ball when punted?

**Ready for the answer?**

a) 3 seconds

b) 19 meters

c) ball was punted at 1 meters high

t=0

y=-2(t-3)2+19

y=-2(0-3)2+19

y=-2(3)2+19

y=-2(9)+19

y=-18+19

**y=1**

Showing you all the steps to graph from factored form -> y=a(x-r)(x-s)

## REFLECTION

## Graphing from Vertex Form the equation given was y=-2(x-3)2+5 and it is in vertex form (y=a(x-h)2+k). h and k are the x and y values of the vertex. so from the equation, the vertex was at (3,5). since there was an 'a' value over 1 (2), it had to be multiplied by the step pattern; which made -2, -8, -18... this showed how the parabola is drawn. | ## Stating Transformations & Relations it is important to know how transformations of a graph occurs with proper terminology. this question asked how the base graph transitioned to become y=-2(x-3)2+5. 1. it reflects the x-axis (negative) 2. vertical stretch by a factor of 2 (a) 3. horizontal shift - right three units (h) 4. vertical shift - up five units (k) _____________________________
you can figure out whether the relation is linear if there is equal first differences. you can figure out whether the relation is quadratic if there is equal second differences. | ## Possible Equations Depending on the Number of x-intercepts1. · a vertex that is below the x-axis and opens up · a vertex that is above the x-axis and opens down 2. · a vertex is on the x-axis and opens up · a vertex is on the x-axis and opens down 3. · a vertex above the x-axis and opens up · a vertex below the x-axis and opens down |

## Graphing from Vertex Form

**y=-2(x-3)**2

**+5**and it is in vertex form (

**y=a(x-h)**2

**+k**). h and k are the x and y values of the vertex. so from the equation, the vertex was at (3,5). since there was an 'a' value over 1 (2), it had to be multiplied by the step pattern; which made -2, -8, -18... this showed how the parabola is drawn.

## Stating Transformations & Relations

1. it reflects the x-axis (negative)

2. vertical stretch by a factor of 2 (a)

3. horizontal shift - right three units (h)

4. vertical shift - up five units (k)

_____________________________

you can figure out whether the relation is linear if there is equal first differences. you can figure out whether the relation is quadratic if there is equal second differences.

## Possible Equations Depending on the Number of x-intercepts

1. __2 x-intercepts__ could be possible if it either has:

· a vertex that is below the x-axis and opens up

· a vertex that is above the x-axis and opens down

2. __1 x-intercept__ could be possible if it either has:

· a vertex is on the x-axis and opens up

· a vertex is on the x-axis and opens down

3. __no x-intercept__ could be possible if it either has:

· a vertex above the x-axis and opens up

· a vertex below the x-axis and opens down

## Expanding you would have to multiply the number outside of the bracket to all the numbers inside of the bracket. then collect like terms to get your final answer. | ## Binomials this is a bit more complex as it has two brackets now. but all you have to do is multiply the numbers in the first bracket to all the numbers in the second bracket. then collect like terms to get your final answer. |

## Expanding

## Factoring!

1. common factoring

2. simple trinomial

3. complex trinomial

4. factoring by grouping

5. perfect squares

6. difference of squares

## 1. Common Factoring

**Factors **- numbers that are multiplied to get a certain number (common factors of 6 is 1, 2, 3, 6 because 1x6=6 & 2x3=6)

say you had to common factor the equation** 2x+4**. 2 is common with both values. so, you would divide 2 out from that equation and put what you factored out (2) in the middle of your new equation along with the divided factor; so the new equation would look like =2(x+4)

another example is **x**2**+6x**. x is common with both values. so, you would divide x out from that equation and put what you factored out (x) in the middle of your new equation along with the divided factor; so the new equation would look like =x(x+6)

one last one would be **2x**3**+8x**2. if you are dealing with powers in both terms, take the lowest power so factor out. so, 2x2 is common with both values. so, you would divide 2x2 out from that equation and put what you factored out (2x2) in the middle of your new equation along with the divided factor; so the new equation would look like =2x2(x+4)

## 2. Factoring Simple Trinomials

When factoring it, we are trying to break it up into two things that multiply up to make it (two bracket). You are looking for two numbers that multiply to get the last term, and are the sum of the second term. You may want to do some rough work on the side and list all the possible numbers that would be the product of the last term. Then, look at which ones add up to your second term. After you figure out what numbers to use, use the guess and check method in your two brackets. To check if your answer is right, just expand to get the original equation.

## Try this out!

**p**2

**+7p+12**

b)** r**2**-10r+16**

c) **w**2**+2w-48**

d) **x**2**-3x-18**

**ready for the answer?**

a) **p**2**+7p+12** =(p+3) (p+4)

b) **r**2**-10r+16** =(r-2) (r-8)

c) **w**2**+2w-48** =(w-6) (w+8)

d) **x**2**-3x-18** =(x+3) (x-6)

## 3. Factoring Complex Trinomials

a complex trinomial is a trinomial which starts with a coefficient thats not equal to 1.

the first thing to look for in a complex trinomial is if you can change it from a complex to a simple trinomial. this can be done by common factoring it. for example: **3x**2**-6x+9**. it could be common factored by 3 and be changed into =**3(x**2**-2x+3) **and thus changed into a simple trinomial.

but you cant always do that, and if you cant then you would have to do another method; which is guess and check. on the side, list all the possible numbers that would be the product of the first term and the last term. then plug them in in two brackets and check your answer by expanding & collecting like terms to get the original equation.

## Try this out!

a) **4x**2**+25x-21**

b) **2x**2**+11x+15**

**ready for the answer?**

a)** 4x**2**+25x-21** =(4x-3)(x+7)

b) **2x**2**+11x+15** =(2x+5)(x+3)

## 4. Perfect Squares

perfect squares are one of the two special factoring cases.

- you know its a perfect square if you can square root a & c
- check by multiplying answer by 2, which should equal b

## Try this out!

**x**2**+8x+16**

ready for the answer?

**x**2**+8x+16**

=(x+4)(x+4)

**=(x+4)**2

__check:__

(x+4)(2)

=8x

## 5. Difference of Squares

difference of squares is the other special factoring case.

- you know its a difference of squares if it is 2 terms, and you can square root both
- the plus and minus is what makes the minus in the first equation

## Try this out!

**x**2**-9**

ready for the answer?

**x**2**-9**

=(x+3)(x-3)

## need extra practice? try this out!

## Reflection

## Completing The Square

There are 4 steps to completing the square. An important formula to remember is (b/2)squared. __Steps:__

1. remove the common factor from the **x**2 and the x-term coefficient

2. Find the constant that must be added and subtracted to create a perfect square. Rewrite the expression by adding and subtracting that constant. **(b/2)**2

3. Group the three terms that form the perfect square. (move the subtracted value outside the bracket by multiplying it by the common factor first)

4. factor the perfect square and collect like terms

*(example on the picture to the right)*

## Solving quadratic equations from standard form (using the quadratic formula)

We already seen some stuff that we could do with standard form, but it involves converting the equation to vertex form. the quadratic formula would allow you to work straight from standard form.

if you have a standard form quadratic equation that equals zero (has an a, b, & c value) (**0=ax**2**+bx+c**), then the formula looks like** x=negative b plus/minus square root of b squared minus 4ac, all divided my 2a**. the plus/minus will give the two solutions.

for example, find the x-intercepts of **y=2x**2**+3x-5**

the a value is 2 from **2x**2

the b value is 3 from **3x**

the c value is -5 from **-5**

now plug in all the values into the quadratic formula!

*(work shown on the picture to the right)*

after you get to the part where you square root your number, the plus and minus indicates that there will be 2 solutions.

**discriminant**- the value in the square root

- if you have a negative discriminant, it is not possible because you cannot square root a negative number. therefore there would be no solutions (x-intercepts).
- if you have a positive discriminant, there would be two solutions (x-intercepts).
- if you ave a discriminant that is equal to zero, there is only one solution (x-intercept).