Grade 10 Quadratic Relations

By: Karanveer Sahota

In The Real World

Quadratics are all around us, we see them almost everyday weather it be outside of in your own home. Some real world examples of quadratics are roller coasters drops, some are in amazing well known towers like The Eiffel Tower in France as well as bridges like The Golden Gate Bridge located in San Francisco and even nature models parabolas in rainbows!

Helpful Notes

Things you should know are..


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=__.

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First & Second Differences

Quadratic relationships consist of second differences not first.
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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=-2(x+2)²-3

  • The a or -2 tells us the vertical stretch by factor of 2
  • The h or +2 tells is the horizontal translation of 2 units left
  • The k or -3 tells us the vertical translation down 3 units


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+2)²-3 would be -3


Transformation:

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


  1. Vertical Translation: The -3 in the expression shows the vertical translation. The negative indicates going downwards and the 3 indicates the units
  2. 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.
  3. Vertical Stretch: The first 2 in the expression tell us the vertical stretch
  4. 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.

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Graphing Quadratics Using Step Patterns

Factored Form

This is another form of quadratic relationships which is written as 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


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


  • So now knowing what the value is (-30), we can figure out the vertex which is (3, -30)
  • Graphed as below
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  • As seen on the graph the x-intercepts are (-2,0) and (8,0) along with the vertex being (3, -30)
Optimal Value (Factored Form)
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

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


5x² -7x+2=0

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)

Axis of Symmetry (Standard Form)

Factoring to turn to factored form

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 is factored


Difference of Squares:

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


16x²-25

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

Factoring difference of squares

Complex Trinomials:

  • The formula for this is a²+2ab+b² an example is 6x²+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
Factoring Complex Trinomials.mp4
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).
Factoring polynomials 1

Perfect Squares:

  • Makes expanding equation simpler
Factoring Perfect Square Trinomials - Ex 2

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

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 above formula in factored form
h=-5t² +20t +60

h= -5(t² -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.

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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
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  • 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

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