# 2.1 Linear functions  (Page 4/17)

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Are the units for slope always ?

Yes. Think of the units as the change of output value for each unit of change in input value. An example of slope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output per units for the input.

## Calculate slope

The slope, or rate of change, of a function $m$ can be calculated according to the following:

where ${x}_{1}$ and ${x}_{2}$ are input values, ${y}_{1}$ and ${y}_{2}$ are output values.

Given two points from a linear function, calculate and interpret the slope.

1. Determine the units for output and input values.
2. Calculate the change of output values and change of input values.
3. Interpret the slope as the change in output values per unit of the input value.

## Finding the slope of a linear function

If $f\left(x\right)$ is a linear function, and $\left(3,-2\right)$ and $\left(8,1\right)$ are points on the line, find the slope. Is this function increasing or decreasing?

The coordinate pairs are $\left(3,-2\right)$ and $\left(8,1\right).$ To find the rate of change, we divide the change in output by the change in input.

We could also write the slope as $m=0.6.$ The function is increasing because $m>0.$

If $f\left(x\right)$ is a linear function, and and are points on the line, find the slope. Is this function increasing or decreasing?

$m=\frac{4-3}{0-2}=\frac{1}{-2}=-\frac{1}{2}$ ; decreasing because $m<0.$

## Finding the population change from a linear function

The population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year if we assume the change was constant from 2008 to 2012.

The rate of change relates the change in population to the change in time. The population increased by $27,800-23,400=4400$ people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.

So the population increased by 1,100 people per year.

The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the change of population per year if we assume the change was constant from 2009 to 2012.

## Writing the point-slope form of a linear equation

Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. Here, we will learn another way to write a linear function, the point-slope form    .

$y-{y}_{1}=m\left(x-{x}_{1}\right)$

The point-slope form is derived from the slope formula.

Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. We can move from one form to another using basic algebra. For example, suppose we are given an equation in point-slope form, $y-4=-\frac{1}{2}\left(x-6\right)$ . We can convert it to the slope-intercept form as shown.

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?

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