Welcome to a lesson
on determining if a table of values
represents a linear function.
Let's begin by talking about the rate of change
between two points.
Given any two points x sub one comma y sub one,
and x sub two comma y sub two, the rate of change
between the points on the interval x sub one to x sub two
is determined by computing the following ratio.
The rate of change is equal to, in general,
the change in output divided by the change in input.
Where if we have x and y coordinates,
we can say the rate of change is equal to the change of y
divided by the change of x.
Or we can use this Greek letter delta to represent change.
So to find the change of y,
we find the difference of the y coordinates,
which would be y sub two minus y sub one.
And to determine the change of x,
we determine the difference of the x coordinates,
which would be x sub two minus x sub one.
But the variables won't always be x and y
so it's important to recognize the inputs
would be the first values in the ordered pairs
and the outputs would be the second values
in the ordered pairs.
Now that we know how to find the rate of change
between two points, if a function is linear,
then the rate of change will be the same
between any pair of points.
This constant rate of change is the slope
of the linear function.
So looking at example one, we're asked to determine
if the function given by the table is linear
by computing the rate of change
between several pairs of points.
If it is linear, give the slope.
And to keep things organized,
we're going to use a table like this
where the first row is going to be the change in output,
or the change in y.
The second row will be the change in input,
or the change of x.
And the final row will be for the rate of change
which is, again, the change in the output
divided by the change in the input.
So we'll use this first column
to determine the rate of change
given by this ordered pair and this ordered pair,
or this point and this point.
And then we'll use this column
to determine the rate of change
between this point and this point.
We'll use the third column to determine
the rate of change between this point and this point.
And finally use this last column
to determine the rate of change
between this point and this point.
Let's do this on the next slide
where I've already worked some of this out.
So we'll first focus on these two points
given by these two ordered pairs.
So let's first determine the change of y,
which is given here.
The change of y is going to be negative 38
minus negative 74 which equals 36.
The change of x is going to be two minus negative one,
which equals positive three.
And therefore, the rate of change
between these first two points is equal to
the change of y divided by the change of x,
which is 36 divided by three which equals 12.
Now if we continue on, if the rate of change
in this last row is always 12,
then we'll say this table is a linear function.
If we get a rate of change that's not 12,
then the table would not be a linear function.
So for the next column, we'll determine the rate of change
using these two points, so these two ordered pairs,
and we'll place the information here.
So the change of y is 34 minus negative 38, which is 72.
The change of x is eight minus two which equals six.
And therefore, the rate of change is 72 divided by six,
which equals 12.
So far so good, in the next column, we use these two points
and determine the rate of change.
And I've already shown this here,
notice how the change of y is 48, the change of x is four,
the rate of change is 12.
And then finally, for the last column,
we'll use these two points, so these two ordered pairs.
Again, it's already here.
The change of y is equal to 132,
the change of x is equal to 11,
the rate of change is equal to 132 divided by 11,
which is 12.
So notice how, looking at the table, the rates of change
between several of the points is 12.
And therefore, the rate of change is the same,
so we have a linear function.
So we have a linear function
and the slope equals 12.
To check to see if we do have a linear function graphically,
we could plot these points in the Cartesian plane,
which I've already done to save time.
So here are the five points on the Cartesian plane
and notice how we could graph a line
that would pass through all five points,
which is the graphical verification
we do have a linear function.
Now the next two examples, we'll do exactly the same way.
The only difference is, notice how the input is n
and the output is T[n].
And so in our table,
the change in output is the change in T[n],
the change in input is equal to the change in n,
and we still have the same ratio
to determine the rate of change.
So using the first two points in the table,
the change in the output or the change of T[n]
is equal to 29 minus two, which equals 27.
The change in n is equal to negative two minus negative five
which is positive three.
And therefore the rate of change
between these first two points is 27 divided by three
which equals nine.
Next we'll determine the rate of change
using these two points, so these two ordered pairs.
So the change in the output, or the change in T[n],
is 114 minus 29 which equals 85.
The change in the input, the change of n is equal to
three minus negative two, which equals five.
Notice how between these two points,
the rate of change is 85 divided by five which equals 17.
So we can actually stop here because
notice how the rates of change are not the same here
and therefore, this is not a linear function.
If we did want to continue, I've also provided that work here.
Using these two points, the change in T[n] is 23,
the change in n is one, and therefore the rate of change
is equal to 23.
Using these two points,
the change of T[n] is 385,
the change in n is 11, the rate of change is 35.
Notice how for this function,
the rate of change is increasing
as the inputs, or n increases.
So this is not a linear function.
If we did look at this graphically,
by plotting the points on the Cartesian plane,
it would look like this.
Notice how we cannot sketch a line
that would pass through all five points,
which is another verification this is not a linear function.
Now let's look at our last example.
Table c, notice how the input is x and the output is H[x].
So we determine the rate of change between these two points.
So the change in the output is equal to the change in H[x],
which is 34 minus 90, which equals negative 56.
The change in the input would be the change in x,
which is three minus negative four, which is seven.
So the rate of change is negative 56 divided by seven,
which equals negative eight.
Continuing, we now determine the rate of change
between these two points, which I've already done here.
Notice how the change in H[x] is negative 16,
the change in x is two,
the rate of change is negative eight.
So far the rate of change is the same.
Using these two points,
the change in the output,
or the change of H[x] equals negative 24.
The change of x is three,
the rate of change is negative eight.
And then finally, using these two points,
the change in H[x] is negative 24, the change in x is three,
and the rate of change again, is negative eight.
So negative 24, three, and negative eight.
So because the rates of change are the same
for several pairs of points in the table,
we can assume the function is linear.
So we have a linear function.
And the slope is negative eight.
To verify this graphically,
we can plot these points in the Cartesian plane,
which I've done here.
And notice how it is possible to graph a line
that would pass through all five points,
which verifies the table does represent a linear function.
I hope you found this helpful.