Welcome to Introduction
to Linear Functions and Slope.
In this lesson, we'll define a linear function
and also determine the slope of a line given two points.
A linear function is a function that fits the form
f of x equals mx plus b
or f of x equals b plus mx
where m is called the slope of the line
and b is equal to the vertical intercept.
We can also say a linear function is a function
that can be graphically represented by a straight line
on the rectangular coordinate plane.
Notice how here we have the graph
of several linear functions.
Now let's talk about the behavior of linear functions.
If as the inputs increase, the outputs also increase.
We have a increase in linear function which means
the line will be going up hill from left to right.
Notice here we have the graph of several increase
in the near functions because the lines are going up hill
from left to right even though some of them
are increasing faster than others.
All of these here functions are increasing.
If as the inputs increase, the outputs decrease.
We have a decreasing linear function which means
the lines will be going down hill
from left to right which we see pictured here.
All these lines are decreasing even though
some of the functions are decreasing faster than others.
If as the inputs increased, the outputs remain constant,
we have a constant linear function which results
on a horizontal line, here we have the graph
of several constant functions.
Notice as the inputs increase, the outputs remain constant.
Then finally we have vertical lines,
I graph these in red because the vertical lines
fill the vertical line test and therefore
vertical lines are not linear functions.
We will be studying vertical lines in this lesson
but just remember vertical lines are not linear functions.
When working with linear functions, we need a way
to describe whether a function is increasing or decreasing
and also how fast the function is increasing or decreasing
and for this we use what's calked slope
were slope is defined as the ratio of the change
in output to the change in input which sometimes we'll see
written this way where this delta means changed.
So again, this slope is equal to the ratio
of the change in output to the change in input.
If we have a linear function in terms of X and Y,
which we often do, we can define the slope as the ratio
of the change of Y to the change of X.
Then finally, if we're given two points
on one of your functions we can determine
the slope but determining the ratio of the change
of the Y coordinates to the change of the X coordinates.
Going back to our graphs again,
if the slope is positive, we have an increase
in the nearer function and the larger the positive slope,
the faster the function increases for example,
linear function with a slope of five would increase faster
or go uphill faster then a linear function
with a slope of positive one half.
If the slope is negative, we have a decreasing
linear function and the smaller the negative slope
the faster the function decreases or goes downhill
for example linear function with a slope
of negative five would go down hill faster,
decrease faster than a linear function with a slope of,
lets say, negative one half.
If the slope is zero, we have a constant linear function,
meaning we have the graph of a horizontal line
and if the slope is undefined because we have division
by zero, we have a vertical line, which again,
we know is not a linear function.
So if we have a linear function like we do graphed here
in blue and we're given two points on the linear function,
we can determine the slope but determining the ratio
of the difference in the Y coordinates
to the difference in the X coordinates.
This ratio gives us the ratio of the change
in output to the change in input.
Let's look at some of examples.
We're asked to determined the slope of the line
containing the given two points
and noticed how I already graphed the two points
on the coordinate plane and sketched the line in blue.
Well here's the point negative two comma three
and here's the point four comma negative one.
Again, because the line is going down hill
from left to right we know we have a decrease
in the near function and therefore
the slope will be negative.
So now we'll find the slope using our slope formula here.
To keep things organized, let's label the ordered pairs.
Let's call this x sub one, y sub one,
and x sub two, y sub two.
Before our next example,
let's have y sub two minus y sub one
that would be negative one minus three
and then we'd have x sub two minus x sub one,
that would be four minus negative two.
Now simplifying,
negative one minus three is negative four and four minus
negative two is the same as four plus two or six.
So our slope is negative four six.
We do have a common factor of two here
so there's that simplified.
This simplified slope is equal to negative two thirds.
Notice how here we had a slope of negative four six
which means when the change in the output is negative four,
the change in the input is positive six.
So notice how if you wanted to travel from this point
on the left to the point on the right, we would have to go
down four units, which is the reason why the change
in the output was negative four and then we'd have to go
right six units which is the reason why the change
in the input was positive six.
Also notice how when we simplified,
we had negative two thirds so notice how from this point
if we were to go instead of four units
we went down two units q
and then right three units.
We still find another point that would be on the same line.
For example, B we have the same question,
find the slope of the line,
passing through the given two points.
Again, I've already plot the two points
and sketched the line, we're here, we have
coordinates negative three, negative one,
and here we have four comma two.
Notice how the line is going up hill from left to right
which means a linear function is increasing
and therefore the slope will be positive.
Let's label our ordered pairs,
x sub one, y sub one,
x sub two, y sub two
so the slope m is equal to
y sub two minus y sub one,
that's two minus negative one and then we have x sub two
minus x sub y that would be four minus negative three.
Simplifying, this become two plus one or three
and here we have four minus negative three or four
plus three which is seven.
This doesn't simplify and therefore
our slope is three sevenths.
So this tells us that when the change in the output
is positive three, the change
in the input is positive seven.
Looking at the graph, if you wanted to travel
from this point in the left to the point on the right,
notice how we'd have to go up three units,
which is why the change in the output is positive three
and then right seven units which is reason
why the change in the input is positive seven.
For C, again I've already plotted the two points.
Here's the point three comma two
and here's the point negative one comma two.
Notice how we have an horizontal line
and therefore they stay constant linear function.
Determining the slope, we have x sub one, y sub one,
x sub two, y sub two.
So we have m slope is equal
to y sub two minus y sub one,
that's two minus two and then x sub two minus x sub one
that'd be negative one minus three.
So two minus two is zero, negative one minus three
is negative four and zero divided by negative four is zero
which you might recognize from our notes.
Anywhere on the line will have a slope of zero.
For our last example,
we have the point two comma negative three
and we have the point two comma one.
Notice how this is a vertical line which means
this is not a linear function and now let's try to find
our slope x sub one, y sub one,
x sub two, y sub two, so the slope m
is equal to y sub two minus y sub one.
That's one minus negative three
and then we have x sub two minus x sub one,
that's two minus two.
Simplifying we have one minus negative three
becomes one plus three or four and notice here
we have two developed by a two in our denominator
which is zero, division by zero is undefined
so we say the slope is undefined.
Sometimes you'll see some sources say there is no slope
but we'll describe it as being undefined.
Remember whenever we have a vertical line,
this slope will always be undefined
because we always have division by zero
because the change in the input is always zero.
I hope you found this helpful.