Sketching linear graphs

Sketching linear graphs using the gradient and y-intercept:

In linear equations in the form y = mx + b
m is the gradient
b is the y intercept

Sketching linear graphs that pass through the origin:

Graphs that pass through the origin are in the form
y = mx
Here, the y intercept is the origin (0,0)
m is the gradient

Sketching parallel lines:

Two lines with the same gradient are parallel lines.
Conversely, parallel lines have the same gradient.

Sketching perpendicular lines:

The products of the gradients of two perpendicular lines is equal to -1.
Conversely, if the products of the gradients of two lines is -1, then the lines are perpendicular.

Sketching linear graphs using x and y intercepts:

To find the x-intercept, substitute y=0 into the equation.
To find the y-intercept, substitute x=0 into the equation.

Sketching vertical and horizontal graphs:

Equations of the form y=b are straight lines parallel to the x-axis (horizontal lines)

Equations of the form x=a are straight lines parallel to the y-axis (vertical lines).


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Gradient is the measure of a line’s steepness (slope).

The gradient of a line describes the change in y values as the x value increases by 1.

The size of the gradient tells us how steep a line is.
The sign ( + or – ) tells us whether the line slopes up or down.

Gradient formula:
Gradient = vertical rise / horizontal run
m = rise / run
= change in y value / change in x value
= y2 – y1 / x2 – x1

The gradient of any horizontal line is zero.
The gradient of any vertical line is undefined.


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Coordinate Geometry

On a Cartesian plane, the horizontal axis is called the x axis and the vertical axis is called the y axis.

The point of intersection of the x and y axes is called the origin. The coordinates of the origin are (0,0).

All the points on the Cartesian plane can be represented by an ordered pair called coordinates which is written as (x,y) and is measured from the origin.

An interval is the line segment formed when two points are joined.

The Distance formula:
The distance d between any two Cartesian coordinates (x1,y1) and (x2,y2) is:

d = square root of [ (x2-x1)squared + (y2-y1)squared ]

The Midpoint formula:
M(x,y)= [ x1+x2/2 , y1+y2/2 ]

The midpoint M(x,y) between two points (x1,y1) and (x2,y2) is:

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Linear Equations

There is a Linear relationship between two variables if a uniform change in the independent variable produces a uniform change in the dependent variable.

A linear relationship produces a linear graph (straight line).

A linear graph is represented by the equation
y = mx + b

Solving a Linear Equation:
A linear equation is a linear relationship where the dependent variable has a specific value. We can solve linear equations by finding the value of the independent variable.

A linear equation has only one solution.

A linear equation can be solved using equivalence.
Equivalence means the left hand side of the equation is equal to the right hand side. So we must do the same operation on both sides to maintain the equality of the equation.


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1. Is burning a chemical reaction?
2. What happens when substances burn?
3. What are oxidation reactions?
4. Is rusting an oxidation reaction? Why?
5. Rusting is a very slow type of burning reaction. True or false.
6. What are fossil fuels? Give examples.
7. When a fossil fuel burns, what reaction takes place?
8. What is an oxyacetylene torch?
9. Burning is also known as _______.
10. Explain how a car combustion engine works.
11. How do rocket fuels help to put a space shuttle in orbit?
12. Which are the liquid and solid fuels used in a rocket?
13. What is a backdraught? Why is it dangerous to firefighters?
14. How does a soda-acid fire extinguisher work?
15. What are methane, ethanol, butane and propane used for?


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