Mr F

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Mr Fs Maths Notes

• Graphs
• 2. Quadratics and Cubics

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2. Quadratics and Cubics
1. What does the Equation of a Curve actually
mean? The equation of a curve, whether it be a
quadratic, a cubic, or anything else, is just a
way of expressing the relationship between the x
co-ordinates and the y co-ordinates that lie on
that curve. Example y x2 3x - 9 This
says that the relationship between all the x
co-ordinates and all the y co-ordinates is get
your x co-ordinate, square it, add on three lots
of your x co-ordinate, subtract 9, and you get
your y co-ordinate SoIf a pair of co-ordinates
has this relationship such as (2, 1) then its
on the curve If it doesnt such as (5, 4) then
it does not lie on the curve What you end up
with is just a curve that goes through all the
co-ordinates which share that relationship
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2. Drawing Curves from their Equation The method
is identical to how we drew straight lines 1.
Choose a sensible value of x one that is small
enough to fit on the paper, and easy enough for
you to work out 2. Carefully substitute it into
the equation to get your y value 3. Do this
enough times to see the shape of the curve 4.
Join them up with a smooth curve (dont have any
sharp, pointy bits) Crucial You are more likely
to get the shape of the curve right if you have a
good knowledge of what shapes different equations
make! Have a quick read though 3. Shapes of
Graphs before you carry on! Number 1 Classic
Mistake People Make Messing up their negative
numbers you must be very careful when
substituting negative xs, whether you are doing
this on a calculator or in your head (see the
next 2 sections) One Final Top Tip Pick x 0 as
one of your points, as it is often nice and easy
to work out the y value!
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• 3. Substituting Numbers in your Head
• If you are asked to draw a curve on a
  non-calculator paper, then you will need to be
  very careful
• Things to remember
• 1. What order you must do operations remember
  BODMAS??
• 2. All you rules of negative numbers!
• Example
• If I was trying to substitute x -2 into y x2
  4x 2, then this is what I would be saying to
  myself in my head
• Okay, lets deal with the squared term first
• (-2)2 is equal to 4, because when you square a
  negative you get a positive
• Next up is 4x which is 4 multiplied by x
• Which is 4 x (-2)
• Which is equal to -8
• So, I have 4 - -8 2
• Well, those two minuses are touching, so they
  become a plus
• So I have 4 8 2

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4. Substituting Numbers using a
Calculator Whilst having a calculator makes
doing tricky sums much easier, it also means you
are likely to get much more difficult numbers to
work with, and if you are not careful,
calculators can do some daft things! Things to
remember 1. Always put your negative numbers in
brackets 2. Always do each calculation twice to
make sure you didnt press a wrong button!
Example If I was trying to substitute x -4
into y x3 2x2 6x 2, then this is the
order I would press the buttons
And if you do all that, you should get a y value
of -6
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5. Using Curves to Solve Equations Seeing as you
have taken all that time drawing a beautiful
curve, you may as well use it to solve an
equation Method 1. If it isnt already,
re-arrange the equation so all the letters are on
the left, and there is either a number or a zero
on the right hand side 2. Draw the graph of the
left hand side of the equation 3. On your graph,
draw a horizontal line through whatever number
was on the right hand side of your equation 4.
Mark on the points where this horizontal line
crosses your curve 5. The x co-ordinates of these
points are the solutions to the equation Note
If there is a zero on the right hand side of the
equation, you are just looking for the points
where the curve crosses the x axis! 6. Putting
it all Together What follows now are three
examples of drawing graphs and then using them to
solve equations I suggest you make sure you can
get each of the numbers in the table yourself
both in your head and on a calculator!
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Example 1
x -2 -1 0 1 2 3 4 5
y 6 0 -4 -6 -6 -4 0 6
Use the graph to solve
We are looking for where the curve crosses the x
axis, which gives us solutions of x -1
and x 4
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Example 2
x -4 -3 -2 -1 0 1 2 3 4
y -27 2 13 12 5 -2 -3 8 37
Use the graph to solve
Again, we are looking for where the curve crosses
the x axis, which gives us solutions of x -3.1
x 0.7 and x 2.5 these are
only rough answers, but that doesnt matter!
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Example 3
x -2 -1 0 1 2 3 4
y 18 7 0 -3 -2 3 12
Use the graph to solve
We must draw in the line y 4 and read off the x
co-ordinates of where the line hits the curve x
-0.7 and x 3.2
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• Good luck with your revision!