Professor George

Good morning class! My name is professor George Khnouf and I will be teaching you math today.  The topic that we will cover today is Linear Functions.  Linear functions are one of the most important and most common types of function that we use in our daily lives.  We use linear functions in many everyday functions such as: calculating the travel times of certain journeys, weights, measurements, interest rates and many many more!

The mathematical definition of a Linear Function is that Linear Functions are functions where the rate of change at every interval is the same (same gradient between every interval), which results in a graph with a straight line like the ones below:

Graph 1:

     

 

Graph 1 is a Linear Equation graph with a positive gradient.

Graph 2:

 

Graph 2 is a Linear Equation graph with a negative gradient. 

- The General Formula for a Linear Function is y=mx+b

m: Gradient

b: y intercept, which will have the coordinate of (0,y)

- Rules to remember when dealing with Linear Functions:

If 2 Linear Functions have the same gradient but difference y-intercepts then these 2 graphs and parallel.

If the gradient of one graph is 2 for example and the gradient of another graph is – 0.5 then these two graphs are perpendicular.  When the there are 2 Linear Functions and one of the Function’s gradients is the negative reciprocal of another then these 2 linear functions are perpendicular.

- If the general formula for a linear function is y=b then it is a horizontal line

- If the general formula for a linear function is x=a then the slope is undefined and its vertical.

- if you want to find the x intercept of a linear function from its equation you replace y with 0 in the equation and solve for x.  If you find to find the y intercept of a linear function from its equation you replace x with 0 in the equation and solve for you.   

In order to find the equation of a Linear Function you must have the gradient and the y intercept.   

-Finding the Gradient:

The gradient is found through:

y2-y1/x2-x1  Or in other words rise/run 

In order to find the Y intercept you need to have first found the gradient and you must have one point that lies on the graph of the Linear Function (x,y) and substitute them all into the equation in order to find b ( the y-intercept)

-Ways in which you might be asked to find the equation of a Linear Function:

There are three main ways in which you might be asked to find the equation of a Linear Function:

1- you could be given a graph such as the one below and be asked to write an equation for the linear function:

 

The Y-intercept is found from the graph (0,1)

In order to find the gradient you take any 2 points from the graph and calculate y2-y1/x2-x1   which equals 2

So the equation of this linear function is y=2x+1

2- you could be given 2 points and be asked to find the equation of the function these two points belong to.

For example, take the points (3,2) and (5,3)

First thing that must be done is finding the gradient by using y2-y1/x2-x1 .  The gradient will be 2.

After finding the gradient you must take one of the 2 points given in the question along with the gradient and substitute it into the equation of y=mx+b

The y intercept will equal -2 so the equation will be y=2x-2

3- you might be given a table that contains data and be asked to find the equation of the linear function by using data from the table itself

Look at the below table and use it to find the equation of the linear function.  But first, you should make sure that it is linear by making sure that the rate of change between every interval is the same:

Input (x)	Output (y)
5	12
9	14
13	16

 If you calculate the rate of change between every interval you will realize that they are the same and is always 0.5

After finding the gradient, you take any combination of Input and Output and substitute them into the general formula along with the gradient to find the equation of the Linear Function.

(5,12)

12= 0.5x5+b

B=9.5

So the general equation for this linear function is:

Y= 0.5x+9.5

This basically covers almost everything you need to know about Linear Functions, I hope this was clear and easy to understand.