Graphs of Quadratic Equations

A quadratic equation is one with an x2 term. It may also have an x  term and a numerical term.

The general formula for a quadratic is:  ax2 + bx + c  (a, b and c are constants)

When you graph a quadratic equation, the shape of the graph is called a parabola. This is a u-shaped curve.

Above is the graph of y = x2

It passes through the origin. The point with the lowest y-value is called the minimum. This is said to be the turning point of the graph, and the gradient is zero here.

The graph of y = x2 – 2 will pass through the point (0, -2). It cuts through the x-axis at two points. By finding the x-co-ordinates of where the graph cuts through the x-axis, you can solve the quadratic equation. (On the x-axis, y = 0)

Using this technique, we can deduce that there is only one solution to y = x2 , which is x = 0

 The graph of y = -x2 is still a parabola, but it is inverted. Instead of having a minimum point, it has a
maximum, the point where the y-value is the highest. Quadratic equations with negative coefficients
of  x2 can still be solved by looking at the x-axis intercept.

A parabola which does not touch or intercept the x-axis represents a quadratic equation which has no solutions.

Other methods to solve quadratic equations:

Factorisation

Using the Quadratic Formula


To get to grips with quadratic equations even more, visit this quadratic equation grapher

Solving a quadratic equation using the quadratic formula

Solving quadratic equations by factorisation is by far the easiest method. Sometimes, however, the quadratic equation will not factorise, so we must solve it by other means. Probably the second easiest method of solving a quadratic is by using the quadratic formula. All you have to do is put the numbers into the formula, work it through, and the answer will drop out.

For a quadratic:  ax2 + bx + c = 0 , you can find the value of x by using the formula:

Notice that there is a +/- sign in the equation. This means ‘plus or minus.’ This is because the quadratic equation has two solutions. Quadratics have two solutions because for any positive value of x2 there are two possible values of x, a positive value and a negative value. This is because if you multiply two negative numbers together, you get a positive answer. 

Eg: 22 = 2 x 2 = 4   but (-2)2 = -2 x -2 = 4

And it really is that simple. You can also solve quadratic equations which don’t factorise by completing the square.

See also: Quadratic graphs

Summary of "The end of privacy?" by Pete Warren

http://www.ocr.org.uk/download/pp_11_jan/ocr_61627_pp_11_jan_gcse_243102_rb.pdf

Summary of “The end of privacy?”  by Pete Warren.

In “The end of privacy” Pete Warren clearly shows that peoples’ privacy is being invaded by the likes of Google™ Street View, This is a technology whereby peoples’ houses will be photographed by the Google™ people. He also shows that companies such as Google™ are quietly introducing new technologies for social networking  and other programs that let companies know what their clients are doing all the time such as location services. Warren believes that these companies are using the information that they receive to predict their clients’ actions. Warren also highlights that these companies are using this information to produce new ways for clients to be dragged into their mobile advertising campaigns without them even realising.  He also mentions that again, Google™ is allowed to use all of your personal information, whenever you sign up to one of their services. Finally, Warren shows that people are oblivious to the fact that this use of technology could evolve into companies influencing your decisions by the amount of information that they can obtain.

In general, Warren is quite concerned that the evolution of mobile technologies is on the way to taking control of our lives, even if it is unnoticeably. Warren believes that technology is an ominous assault on peoples’ privacy and that many people are oblivious to the amount of technology that is being used to do so.  He thinks that companies which exploit such technology are betraying people  and that privacy campaigners are shocked by this. Despite this, He thinks that such technologies that invade peoples’ privacy have great  potential in the mobile market. However He believes that the greatest betrayal is that companies are making people sign up to their privacy deifying policies which allow them to use all your personal data. Overall, Warren thinks that the mobile industry is a cruel and sinister way for you to be denied of your privacy without even knowing it.

Geography notes: Energy Resources

Geography - Energy Resources

Non-renewable resources = finite, non-sustainable e.g. fossil fuels

Renewable resources = flow of energy: can be used over and over again e.g. solar and geothermal energy

Sustainable: renewable and self-generating however their existence can be threatened when over used, and their value may be reduced

Increasing demand for energy resources:

Population growth leads to an increasing demand of energy resources

Economic development , countries try to raise the quality of life

Increasing wealth

How can we protect the energy resources?

Conservation

Recycling

Glass and waste paper

Greater efficiency

Home insulation, efficient cars

Developing renewable energy resources

Wind, tide, sun

Controlling pollution

Reducing emission

Using appropriate technology

Low cost energy

Non-renewable energy resources

Coal

Advantages

Likely to last for over 300 years

Increase in technology has allowed a larger output per worker

And conversion to energy more efficient

Disadvantages

Most easily accessible deposits used up

With increased competition, its used a lot less

Burning of coal produces air pollution – contributes to global warming

Deep mining is dangerous

Its heavy and bulky to transport

Oil and Natural gas

Advantages

More efficient than coal

Easier to transport than coal

Less harmful to the environment than coal

Gas is cheaper and cleaner than oil

Safer than nuclear

It’s the basis of a huge petrochemical industry

Disadvantages

Reserves will only last another 50-70 years

New fields are difficult to discover and produce

Danger of oil leaks and fires

Releases nitrogen oxide and sulphur dioxide

Vulnerable to political, economic and military pressures

Nuclear energy

Advantages

Only a very small amount of fuel needed

Wont be exhausted for a long time

Risk of accident is minimal

Waste is limited and can be stored underground

Contributes less to greenhouse effect than others

Disadvantages

Not clear how safe it actually is

When a disaster happens, it’s on a huge scale

It is non-renewable

Cannot be used for heating and transport

Potential health risks

Waste can remain radioactive for many years

Cost of decommissioning is very high

Fuelwood

Advantages

Provides basic needs – shelter, food, fuel, shade

No cost

Disadvantages

Wood is running out very quickly

Collecting is a very time consuming job

Desertification

Renewable energy resources

Hydro-electric power

Advantages

Relatively cheap

Reduce the risk of flooding

Only limited pollution

Disadvantages

Dams are very expensive to build

Large areas of wildlife and farmland have been flooded

Visual pollution

Geothermal energy

Advantages

Provides a constant supply of energy

Pollution free

                  Disadvantages

High construction cost

High maintenance cost

Limited to volcanic areas

Threat of eruption and sulphuric gases

Wind

Advantages

It is safe

It is clean

Does not contribute to global warming or acid rain

Minimal effect on ecosystems

Winds are stronger in winter – when demands are highest

Relatively cheap after building the initial turbine

Source of income for farmers

Likely to be built offshore

Disadvantages

Wind does not blow all the time

Electricity cannot be stored

Visually un-appealing

Not very efficient

100000 turbines needed for 20% of Britain’s energy

Noisy

Solar energy

Advantages

Safe

Pollution free

Efficient

Limitless

Disadvantages

Expensive to build solar stations

Hindered by the weather

Potential is great in MEDC’s

Hydrogen

Advantages

Only biproduct is vapour

Has great potential for the future

Disadvantages

Needs electricity to produce

Research costs a lot

Tidal

Advantages

Renewable

Produce sustained, constant electricity

                 Disadvantages

Visually unattractive

Destroys wildlife habitats

Expensive to build

Waves

Advantages

Very high energy levels

Disadvantages

Difficult to design materials to withstand the force of waves

Difficult to convert the energy into electricity

Biomass

Advantages

Very cheap to produce

Can be used to power many things

Disadvantages

Methane is a greenhouse gas

Dung cannot be used as a fertilizer