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
- 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
- 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
- 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 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
- 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
- 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 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
- 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
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
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.
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