# Grade 10 Quadratic Relations

### By: Karanveer Sahota

## In The Real World

## Helpful Notes

**Vertex:** Is the maximum or minimum point on a graph, it is the point where the graph changes direction. Labeling is followed as (x,y).

**Minimum/Maximum Value (Also the Optimal Value):** The highest or lowest value the parabola takes on, which is the highest or lowest point consisting on the y-axis. This is labeled as y=__.

**Axis of Symmetry:** The vertical line which cuts the parabola down the middle, this is labeled as x=__.

**Y-Intercept:** Is where the parabola intercepts the x-axis, labeled as (0, __).

**X-Intercepts: **Where the parabola intercepts the x-axis, it is labeled as (__, 0).

**Zeros:** The x-value which makes the equation equal to zero. Labeled as x=__.

## First & Second Differences

## Vertex Form

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

*²***+k**

tells us the stretch on the parabola**a**tells us the horizontal translation or the x-value**h**tells us the translation of the y-value**k**

Example: **y=-2(x+2)²-3**

- The
*a*or -2 - The
*h*or +2 - The
tells us the vertical translation down 3 units*k*or -3

__Axis of Symmetry (AOS): (Above as well)__

- The AOS is
*x=h**h*in the expression*y=a(x-h)²+k* - The AOS of the parabola above is -2

__Optimal Value: (Above as well)__

- It is written as
*y=k* - The optimal value of
*y=-2(x+2)²-3 would be -3*

__Transformation:__

They can occur in vertical or horizontal, vertical stretches and reflection

**Vertical Translation:**The -3 in the expression shows the vertical translation. The negative indicates going downwards and the 3 indicates the units**Horizontal Translation:**The number inside the bracket, 2 shows the horizontal translation by 2 units. Since the number is positive it will move to the left but if it were negative then the parabola would move right.**Vertical Stretch:**The first 2 in the expression tell us the vertical stretch**Reflection:**The negative sign indicates the vertical reflection, if the number was a positive then the parabola would open upwards, since the number is a negative then the parabola would be flipped upside down.

__X-intercepts or Zeros:__

- You use this when needed to find x-intercept or zeros by setting y=0 and then solving

__The Step Pattern:__

To graph this you must find out what the vertex is, which is (-2,-3). Now that you know the vertex follow the rule of graphing which is over one up one, over two up four. Now using the step pattern since we have a a-value we must multiple it by -2. So instead of going over one up one and over two up four we are going to go over one down two and over two down 8 (1 multiplied by -2 is -2 and 4 multiplied by -2 is -8, since they are negative numbers we go down instead of up). This can be graphed as below.

## Factored Form

*y=a(x-s)(x-t)*and also as

**y=a(x-r)(x-s)**- To graph this form we must find what the x-intercepts are, we also have to add up the two x-intercepts and divide them by two so that we can have the minimum/maximum value.
- Example: y=2(x-8)(x+2) so the x-intercepts are (8,0) and (-2,0), now you add and divide them 8+(-2)/2 =2. So now you would sub in the 3 as x
- This will figure out what the minimum/maximum values are and help us find the vertex

- y=2(x-8)(x+2)
- y=2(3-8)(3+2)
- y=2(-5)(5)
- y= -50

* *

- So now knowing what the value is (-30), we can figure out the vertex which is (3, -30)
- Graphed as below

- As seen on the graph the x-intercepts are (-2,0) and (8,0) along with the vertex being (3, -30)

__Zeros or X-intercepts:__

- Zeros are simply just used by setting the numbers to zero
- Example: y=0.5(x+3)(x-9), must set y to zero (y=0)

y=0.5(x+3)(x-9)

0=0.5(x+3)(x-9)

x+3= 0

x= -3

x-9= 0

x= 9

- The zeros or x-intercepts are -3 and 9

## Standard Form

- The form of this method is
*y=ax²+bx+c*

__Quadratic Formula:__

- This formula is the one you are suppose to use when doing standard form

__Zeroes:__

- Using zeroes all we have to do is put the zero in the formula
- Example:

**a=5**

**b=-7**

**c=2**

=-(-7)±√7² -4(5)(2)/2(5)

= 7±√49-40/10

=7±√9/10

=7±3/10

1. 7+3/10

** =1**

2. 7-3/10

**=0.40**

- First, you indicate what a,b and c is so in this case (a=5, b=-7 and c=2)
- Then, Sub into the equation
- Now, solve the numbers
- After than we get our two equations and x-intercepts

__Optimal Value:__

- For this we need to substitute the AOS with the original equation

__Completing the Square:__

- We use this to turn standard form into vertex form
- First factor them into the vertex form
- Place brackets around the numbers that are left
- Add and subtract the number from the brackets
- Write x and divide b by two then put squared outside of the bracket writing into vertex form

__Axis of Symmetry:__

- The formula for this is (-b/2a)

## Factoring to turn to factored form

__Common Factoring:__

- This is when you need to find out the GCF
- Example:

4(x+2y)

- What I did was divide everything by two, and putting it into brackets which is factored

__Difference of Squares:__

- Squared terms
- Example: x²-9 would be (x-3)(x-3)

16x²-25

(4x+5)(4x-5)

__Complex Trinomials:__

- The formula for this is
*a²+2ab+b² an example is 6x**²+11x+4* - First you would find number that will work with this like

(3x+4)(2x+1)

(3x)(2x)= 6x

(3x)(1)= 3x

(4)(2x)= 8x

(4)(1)= 4

6x²+3x+8x+4

6x²+11x+4

- We got the like terms and put them together to get the answer

__Simple Trinomial:__

- Simple trinomials formula is
*x²+bx+c* - An example is w²+5w+6
- First I need to place w in the brackets for the reason being that we need 2 so that we can get w². Now we look for the multiples of 6 (2x3,6x1), now we look at 5w and check what will give us 5, so (w+3)(w+2).

__Perfect Squares:__

- Makes expanding equation simpler

## Connections

__Vertex to Standard:__Need to expand as well as collect the like terms available

__Standard to Factored:__ Need to factor out as much as you can

__Factor to Vertex:__ Need to expand on equation and collect as many like terms possible

__Vertex to Factored:__ Need to make y to 0 (y=0) and then figure out the x-intercepts

__Factored to Standard: __Need to do collect terms and expand too

__Standard to Vertex:__ Use the completing the square method

## Word Problems

*²*+ 20t+ 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)

- The rock will hit the ground at 6 seconds because since the 6 is negative you would put a positive instead because time cannot be negative

2. Write and Simplify an expression to represent the area of the given composite figure.

(Figure below)

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

=3x+x²-2x+2x-4

=3x+x²-4

**b. If the area of the shape is 36cm², determine the value of x.**

A= x²+3x-4

36= x²+3x-4

0= x²+3x-40

0= (x+8)(x-5)

x= -8 OR x= 5

X=5 is the correct answer because it is not negative.

## Reflection

- When we started this unit I understood it well but when it got harder and harder it got more complicated for me, and i understood some things and had difficulty with others. This was my first time dealing with parabolas. I did good on the first unit test but not so well on the second.
- In this question I was suppose to simplify an expression to represent the area of the figure
- I put a addition sign between the brackets when they were not needed I was in fact suppose to multiply them

The answer:

- = 3x+(x+2)(x-2)
- =3x+x²-2x+2x-4
- =3x+x²-4

- I got the first line right but then did the multiplication wrong
- I added an extra 6t+4 and 4t+4
- The correct way:

*2(3t+2)+ 4(t- 1)(t+1)*

*= 2(3t+2) (3t+2)+ (4t- 4)(t+1)*

*= (6t+4) (3t+2)+ 4t²+ 4t- 4t- 4*

*= 18t²+ 12t+ 12t+ 8+ 4t²- 4*

*= 22t²+ 24t+ 4*