Mr F’s Maths Notes

Graphs

2. Quadratics and Cubics

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”

So…If a pair of co-ordinates has this relationship… such as (2, 1)… then it’s on the curve

If it doesn’t… 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

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 (don’t 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 x’s, whether you are doing this on a calculator or in your head (see the next 2 sections)

One Final Top TipPick x = 0 as one of your points, as it is often nice and easy to work out the y value!

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 remember1. What order you must do operations – remember BODMAS??2. All you rules of negative numbers!

ExampleIf 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, let’s 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…

• Which equals 10

• So, the point I need to plot has the co-ordinates (-2, 10)

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 remember1. Always put your negative numbers in brackets2. Always do each calculation twice to make sure you didn’t press a wrong button!

ExampleIf I was trying to substitute x = -4 into y = x3 + 2x2 – 6x + 2, then this is the order I

would press the buttons:

3x 22x

6x

And if you do all that, you should get a y value of… -6

2

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

Method1. If it isn’t 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 side2. Draw the graph of the left hand side of the equation3. On your graph, draw a horizontal line through whatever number was on the right

hand side of your equation4. Mark on the points where this horizontal line crosses your curve5. 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!

x -2 -1 0 1 2 3 4 5

y 6 0 -4 -6 -6 -4 0 62 3 4y x x

Example 1

2 3 4 0x x 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

x -4 -3 -2 -1 0 1 2 3 4

y -27 2 13 12 5 -2 -3 8 373 8 5y x x

Example 2

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 doesn’t matter!

3 8 5 0x x

x -2 -1 0 1 2 3 4

y 18 7 0 -3 -2 3 1222 5y x x

Example 3

22 5 4x x 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

Good luck with your revision!