# Triganometry, Graphs & Measurements

- Created by: Kathryn
- Created on: 10-05-14 08:41

## Pythagoras Theorem

## Sin, Cos, Tan

## Formula Triangles

## Linear Graphs

Linear functions can be written in the form **y = mx + c** where y and x are variables and m and c are constants (numbers).

If you write them like this then **m** is the **gradient** and **c** is the **y-intercept** (point where it crosses the y-axis). The graphs of linear functions are **straight lines**.

**To find m:**

Pick any two points.

**To find c:**

c is the point where the graph crosses the y-axis.

## Quadratic Graphs

Quadratic functions can be written in the form:

**y = ax ^{2} + bx + c**

where a, b and c are constants and 'a' doesn't equal zero.

Quadratic graphs are always **parabolas** ('U' shapes).

The really important bits of a quadratic are:

**Where it turns (the bottom of the 'U')**

**Where it crosses the x-axis (if it does!)**

The solutions of a quadratic are where the graph crosses the x-axis!

## Cubics and Reciprocals

**You need to be able to:**

- Plot and draw these.
- Recognise the shapes.
- Read the solutions from the graph (cubics only).

*Cubics* can be written in the form:

**y = ax ^{3} + bx^{2} + cx + d**

*Reciprocals* are where the x is on the bottom of a fraction.

Drawing their graphs - **Table - Axes - Plot - Draw - Label**

The *solutions* of a cubic are where it crosses the x-axis and it can have up to 3.

## Graphs of simultaneous equations

As simultaneous equations at GCSE are **linear** (can both eb written in the form y = mx + c) their graphs will be straight lines.

The solution (x-value and y-value) is where the straight lines intersect (cross one another).

## Inequalities - regions on a graph

**To draw a graph:**

- Change the inequality sign to an '=' sign.
- By choosing 4 or 5 different values for x, make a table of co-ordinates.
- Draw and label the line (make it dotted if the inequality sign is < or >).
- Choose a test point (not on the line!).
- Put the x and y values of the test point into the inequality.
- If it works, shade and label that side of the line with the inequality.
- If it doesn't work, shade and label the other side.

## Travel Graphs

*Distance/time*

If you show a graph of a journey showing distance travelled (on the y-axis) against time (on the x-axis):

- The
**gradient**(or slope) of the graph represents the**speed**. - A horizontal section indicates that you have stopped.
- A section sloping up means that you are going away.
- A section sloping down means you are coming back.
- The steeper the line, the faster you are going.

*Speed/time*

- The
**gradient**(or slope) of the graph represents the**acceleration**. - The
**area**under the graph (for any section) is the**distance travelled**(in that section). - A horizontal section indicates constant speed (no acceleration).
- A section sloping up means accelerating.
- A section sloping down means slowing down.
- The steeper the line, the quicker the acceleration.

## Area

## Volume

## Metric

**Length**

1cm = 10mm

1m = 100cm

1km = 1000m

**Mass:**

1kg = 1000g

1 tonne = 1000kg

**Volume:**

1 litre = 1000ml = 1000cm^{3}

**Note to remember:**

1m = 100cm 1m^{2} = 10 000cm^{2} 1m^{3} = 1 000 000cm^{3}

## Imperial

**Here are some more facts that you should know:**

**Length:**

1 foot = 12 inches

1 yard = 3 feet

1 mile = 1760 yards

**Length:**

1 pound = 16 ounces

1 stone = 14 pounds

1 ton = 160 stones (or 2240 pounds)

*Note:* the different spellings of tonne (metric) and ton (imperial).

**Volume:** 1 gallon = 8 pints

## Conversion

#### Kilometres and Miles:

#### Miles to Km - Multiply by 1.6

#### Km to Miles - Multiply by 0.62

#### Kilograms and Pounds:

#### Kg to Pounds - Multiply by 2.2

#### Pounds to Kg - Multiply by 0.45

#### Litres and Gallons:

#### Litres to Gallons - Multiply by 0.22

#### Gallons to Litres - Multiply by 4.55

#### Metres, centimetres, feet and inches:

#### Inch to Cm - Multiply by 2.54

#### Cm to Inch - Multiply by 0.39

## Lotus

A **locus** is simply a set of points that satisfy some sort of condition.

*Distance from a point*

A circle around the point!

*Distance from two points*

**A perpendicular line straight down the middle of the points:**

- Set your compasses so that they are
**roughly**the same as the distance between the points (or less if you don't have a lot of room!). - Put the point of the compasses on the first cross and do two arcs - one above the points and one below.
- Put the point on the second cross and do the same thing so that you cross the first arcs (making sure you keep the compasses the same distance apart).
- Now simply draw a line straight down the middle through the points where the arcs cross.

## Lotus (2)

*Distance from two lines*

**The set of points that are the same distance from two lines is a straight line down the middle which bisects the angle (cuts it in half):**

- Get a pair of compasses and place the point where the two lines meet. Draw little arcs that cross each of the lines.
- Now, keeping the compasses set, put the point on each line where your arc has crossed it and draw another little arc in-between the two lines. You should now have another two little arcs in the middle.
- Draw a straight line from the angle through the point where your little arcs cross and you've done it!

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