### Lets make this easier

Did you know quadratics is all around us. We see them pretty much everyday. Some examples will the drops of roller coaster drops, rainbows, Eiffel tower in France and even some bridges.

## Things you need to know

Vertex: Is the maximum or minimum point on a graph, it is the point where the graph changes direction.

Minimum/Maximum Value: The highest or lowest value on the parabola which is the highest or lowest point on the y-axis. This is labeled as y=__.

Axis of Symmetry: The vertical line which in goes through the centre of the parabola, 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=__.

## Type of equations

There are two types of equations which is vertex form and factored form

## Vertex form

Vertex form equation is y=a(x-h)²+k

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

Example: y=-4(x+6)²-8

• The a or -2 tells us the vertical stretch
• The h or +3 tells is the horizontal translation
• The k or -4 tells us the vertical translation

Axis of Symmetry (AOS): (Above as well)

• The AOS is x=h, so the 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+3)²-4 would be -4

Transformation:

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

1. Vertical Translation: The -4 in the equation above shows the vertical translation. The negative shows that its going to be downwards and the 4 shows the units
2. Horizontal Translation: The number inside the bracket, 3 shows the horizontal translation which is by 3 units. If the number is positive it will move to the left but if the number was a negative then the parabola would move right.
3. Vertical Stretch: The 2 in the equation tell us the vertical stretch
4. Reflection: The negative sign shows the vertical reflection, if the number was a positive then the parabola would open upwards, If the number is a negative then the parabola would be flipped image which is upside down

X-intercepts or Zeros:

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

## y=2(x-4)-3

The Step Pattern:

To graph you need to find what the vertex is, in this case it is (-2,-3). Now that you know the vertex follow the step pattern which is over one up one, over two up four. Now using this we have to multiple the a value by -2. So instead of going over one up one and over two up four we are going to go over one down four 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).

Solving X Intercepts from Vertex Form Grade 10 Academic Lesson 6 5 6 19 14

## Factored form

This is a form of quadratic relationships which is written as y=a(x-s)(x-t)

• To graph the standard form we need to 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=4(x-4)(x+4) so the x-intercepts are (4,0) and (-4,0), now you add and divide them 4+(-4)/2 =2. So now you would sub in the 3 as x
• This will figure out what the minimum/maximum values are and help to find the vertex

1. y=2(x-4)(x+4)
2. y=2(3-4)(3+4)
3. y=2(-1)(7)
4. y= 2(-7)
5. y=-14
• So now knowing what the value is (-14), we can figure out the vertex which is (3, -14)
• Graphed as below

## Zeros or X-intercepts:

• Zeros are 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+6)(x-18)

0=0.5(x+6)(x-18)

x+6= 0

x= -6

x-18= 0

x= 18

• The zeros or x-intercepts are -6 and 18

## Standard Form

The equation for this type of form is y=ax²+bx+c

## Zeroes

• When using zeroes the only thing you have to do is put the zero into the formula
• For example:

5x² -7x+4=0

a=5

b=-7

c=4

=-(-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 find what a,b and c is so in this equation (a=5, b=-7 and c=2)
• After, Sub a,b,c into the equation
• Then, you can solve the equation
• And finally we get our two equations and x-intercepts

## Completing the square

• We use this to turn standard form into vertex form
• First factor them so they are the vertex form
• Place brackets around the numbers that are remaining
• 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)

## Complex Trinomials:

• The formula for this is a²+2ab+b² an example is 6x²+11x+4
• First you would find numbers that work with this for example

(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 find the final solution

## Simple Trinomial:

• Simple trinomials formula is x²+bx+c
• An example is g²+5g+6
• First I needed to place g in the brackets for the reason being that we need 2 g's so that we can get g². Now we look for the multiples of 6 (2x3,6x1), now we look at 5g and check what will give us 5, so (g+3)(g+2).

## Common Factoring:

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

4x+8y

4(x+2y)

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

## Difference of Squares:

• Squared terms
• Example: x²-4 would be (x-2)(x-2)

## Perfect Squares:

• Makes equations simpler when you are expanding

An example would be y=(x+3)²

## Key Things

Vertex to Standard: Expand and gather the like terms

Standard to Factored: Factor out as much as you can

Factor to Vertex: Expand on equation and gather all the 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 gather terms and expand as well

Standard to Vertex: Use the completing the square method

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

=(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.

## Word Problem

1. The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h = -5t² + 20t+ 60, where t is the time in seconds, and h is the height in meters.

a) Write the formula stated above in factored form
h=-5t² +20t +60

h= -5(t² -4 -12)

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

• The rock will hit the ground at 6 seconds because the 6 is negative number you would make the number a positive.