Lets make this easier
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
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
- 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
They can occur in vertical or horizontal, vertical stretches and reflection
- 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
- 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.
- Vertical Stretch: The 2 in the equation tell us the vertical stretch
- 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
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).
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
- y= 2(-7)
- 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)
- The zeros or x-intercepts are -6 and 18
The equation for this type of form is y=ax²+bx+c
- When using zeroes the only thing you have to do is put the zero into the formula
- For example:
- 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)
- 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
- We got the like terms and put them together to find the final solution
- 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).
factoring to turn to factored form
- This is when you need to find out the GCF
- 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)
- Makes equations simpler when you are expanding
An example would be y=(x+3)²
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
b. If the area of the shape is 36cm², determine the value of x.
x= -8 OR x= 5
X=5 is the correct answer because it is not negative.
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.
- When we started this unit I understood it, but it got more complicated for me, and I understood some but had but had difficulty with others. This was my first time I learnt about parabolas.
- I did well on the first quadratic quiz but as the the test came I just wasn't doing so well