Maths revision

Maths - What's so good about it anyway?

• Maths revision and study guide
• Everything you need to know to at very least pass and at very best enjoy
• Includes:
• - Exploring Parabolas
• - Sketching Parabolas with Transformations
• - Sketching Parabolas using Factorisation
• - Sketching by Completing the Square
• - Sketching using Quadratic formula
• - Applications Involving Parabolas
• - Non - Linear Relations
• - Drawing Exponential Functions
• - Exponential Curves

Exploring Parabolas

• All graphs feature:
• -Turning point
• - Axis of symmetry
• - Y-intercept
• - X-intercepts
• A parabola is the graph of a quadratic relation. The basic parabola has the rule Y= x2
• A parabola can be:
• - Dilated = How steep the graph is
• - Reflected = Smily face or sad face
• - Translated = left or right along the axis
• 
• Bildunterschrift: Examples of different parabolas

Parabolas With Transformations

• Transformations in this case means that the parabolas has moved up or down, or left or right. This is determined with in the formula
• Y = a(x-p)2 -q
• a - Determines whether the graph will be happy or sad and it dilution
• x - Does not change it forever remains x
• p - Determines if the graph goes left or right
• q - Determines wether the graph goes up or down
• But how does one sketch the formula?
• Good question, take it step by step
• 
• Bildunterschrift: Let this be our example equation

Steps to a healthy parabolas

• Step 1: Plot the point of p and q - this is now the turning point of the graph - p is x and q is y
• Step 2: Find the y and X intercepts - remember to find Y, x = 0 To find x, Y = 0
• Step 3: From these three draw a curved graph - ensure that the peak of the graph is curved
• Step 4: Find the dilution by going across one and then going down until you reach the graph again

Steps to a healthy Formula

• Finding the rule of a Parabolas
• Step 1: Find the dilution - is the graph happy or sad if sad place a - next to the dilution
• Step 2: Is the graph on the left or the right from the point of (0,0) - if the graph is on the right than p is a negative. If the graph is on the left than q is positive -- This is found by quite simply finding the x value in the turning point
• Step 3: Is the graph ​​​​going up or down from the point (0,0) - If the graph is below (0,0) than it is negative. If it is above than it is positive. -- This is found by quite simply finding the Y value in the turning point
• Step 4: Use the standard formula to create the new formula - Y = a( x- p )2 - q

Parabolas using Factorism

• This is the best way to find x intercepts. You should remember how to factorise an equation after last semesters exams but heres a recap
• Y = x2 -2x -3
• Find two numbers that add to give 2x and when timed together give -3 The form is written as follows y = (x - 4 ) ( x + 1 ) do you remember what happens next? that's right find x by making one y equal to 0 : 0 = x - 4 or 0 = x +1 = x = 4 or x = -1
• The line symmetry can be found by finding the middle of the two x intercepts - the middle of 4 and -1 is 3.5
• There for the line of symmetry is 3.5
• Since the line of symmetry is always on the x axis the value for the line of symmetry is the x value of the turning point
• Now to find the Y value of the turning point substitute our new x value value into the equation and solve for Y

Completing the Square

• Remember how to work it out : ( x b / 2 )2 - ( b / 2 )2 + C - remember the - in the expression never changes
• The best way to show this is to just use an example equation
• The steps are :
• 1 Simplify
• 2 Find the Turning points
• 3 Find the Y intercept
• 
• Bildunterschrift: Use this as an example problem

Steps 1 and 2

• Step 1 - simplify. Use the the rule given in the slide before and rewrite the equation
• Y = X2 + 6x - 2
• Y = ( X + 3x )2 - 9 - 2 - remember the formula for completing the square
• Y = ( X + 3x )2 - 11
• Step 2 - Find the turning points - To do this simply take the C value in this case 11 this if the Y value in the T.P. The X value in the T.P. is in this case three
• Step 3 - Find the Y intercepts - to do this

Applications of Parabolas

• There are 6 steps to ensuring your parabolas has a happy life
• 1 : Define the variables
• 2 : Form the equations
• 3 : Solve the equations for area - if necessary
• 4 : Decide on a suitable range of values for the variables
• 5 : Sketch the graph showing all the key features
• 6 : Find the maximum or minimum turning point
• 7 : Now go back and answer the question
• There are two different ways of doing this so we'll split it up into two parts
• Lets do this !

Part a

• 
• Bildunterschrift: Let this be our problem

Steps 1, 2 and 3

• Define the variables - Perimeter = 100m Length = L Width = x
• Write an equation - 100m = 2L + 2x
• Look back at the question what is it asking? In this case it is asking to write an expression for the length of the paddock. this means we have to rearrange the equation to equal L before we can continue.
• 2L + 2x = 100 L + x = 50 -- This is because we only need one length so halve to make one
• L = 50 - x
• We now need the equation to be for area we have the length so now all we need to do is times by the width which if you recall is x
• A = x ( 50 - x )
• 

Steps 4

• 
• Now we need to decide an appropriate values. As we are looking for the width or x, a negative number or 0 is obviously not going to work. So our number has to be larger than 0. We also know that the value has to be less than 50 because that is our length.
• Here's what we have so far
• x > 0 and 50 -0 > x
• Remember to simplify these pretend the > is an = So the simplified version of this is
• 0 < x < 50
• So now we have suitable variables move on to step 5, sketching graphs

Step 5 and 6

• Plot the x intercepts which we just worked out. Now plot the line of symmetry - remember this is the middle number between the x intercepts in this case its 25
• Substitute the line of symmetry ( 25 ) back into the original equation instead of x
• A = 25 ( 50 - 25)
• A = 25 X 25
• A = 625
• So now we use these two variables to find the turning point. The turning point is ( 25 , 625 )
• Now we have all the variables needed to draw this parabolas
• To find the maximum or minimum turning point find the Y value in the turning point in this case 625
• Remember is it is a maximum point if the graph is a sad face and it is minimum point if the graph is a happy face
• 
• Bildunterschrift: This is what the graph will look like

Step 7

• Go back and make sure you have answered everything required most of the time there will be an extra question so just read through them again because there may be something you missed.
• We can see here there is another question down the bottom that has been added
• The question is:
• What will the dimensions of the paddock to achieve its maximum area?
• We know the maximum area is 625 and from our graph we can see that it hits 25 before coming back down so we can say that it becomes a square and there fore the dimension will be
• Length = 25m Width = 25m
• 
• Bildunterschrift: Question f is added in

Part b

• 
• Bildunterschrift: Let this be our example problem

Steps 1, 2 and 4

• Define the variables - h = height d = horizontal distance
• Write the equation - this is easy because the equation is already given - h = -1/16 ( d - 10 )2 + 9
• Look back at the question, in this case it is asking to sketch the graph of the rule for 0 < d < 22 and to find the coordinates of the turning point
• For step 4 we already have a suitable range between 0 and 22 so we can immediately move on
• 

Step 5

• 
• so now we sketch the graph
• First find the height by solving the equation giving
• h = - 1/16 (d - 10 )2 + 9
• h = 2.75
• Plot 2.75 onto the graph - remember that distance is the horizontal axis and the height is on the vertical axis
• Next plot 0 and 20 onto the d axis -- these are our x intercepts
• Now plot the line symmetry - in this case it is 10 - you can use this as your x value in the turning point
• Now substitute this new x value back into the equation to find the Y value of the turning point - in this case it is 9
• The Y intercept is 2.75 so the height which we have already worked out

Steps 6 and 7

• 
• Find the maximum or minimum turning point -- in the case the graph is frowning so the turning point is maximum
• Now go back and answer the questions remaining in this case the questions are :
• What is the maximum height the javelin reaches ?
• This is the Y value in the turning point so the answer is 9
• What horizontal distance does the javelin travel?
• this can be found on the graph as it is our highest x intercept so in this can the answer is 22

Non - Linear Relations

• These are basically parabolas but they make a circle
• This is the basic formula : X2 + Y2 = r2
• r = radius
• When an equation looks like this: X2 + Y2 = 400 we can work out the centre and the radius. The centre is immediately ( 0,0 ) because the equation has X2 + Y2 and so automatically the centre is ( 0,0 ). The radius is then the square of 400 which is 20

Drawing Graphs Of Exponential Functions

• You should be given a situation and a table of values. You will then have to plot the values on the graph and find the formula.
• So there are three steps:
• Step 1 : Plot the values on the graph
• Step 2 : Draw in the graph
• Step 3 : Find the Formula
• 
• Bildunterschrift: This our table and the example we are going to use

Steps 1, 2 and 3

• Step 1 : plot the values onto the graph - Time goes on the horizontal axis and Cells go on the Vertical axis
• Step 2 : Draw a line connecting the dots - this line should be curved
• Step 3 : Now to determine the formula. There is no rule for these formulas so you just have to find the relationship between time and the cells. in this case the formula is C = 2t Ok so another way you could look at it is x times y. X is the y intercept and the x is the ratio
• 
• Bildunterschrift: Cells increase by splitting in two. One cell splits to become two, thee split to become four and so on.

Probability Unions and Intersections

• Key Words
• - A set - IS a collection of elements that can include letters, numbers or objects
• - The sample space - Is the set of all possible elements considered in a particular situation
• - Venn Diagram - Illustrates how all elements in the sample space are distributed among events
• - Null or empty set - Is a set with no elements
• - Intersection - All elements that belong to both set A and set B
• - Union - All elements that belong to either set A or set B
• - Mutually exclusive - If set A and set B have nothing in common
• - Complement - Neither from set A or set B
• 
• Bildunterschrift: Some symbols for the previous terms