Gustavo got a job selling lightsabers
at a local park and swap.
He is paid a small fixed amount
plus a percent of each item he sells.
The table below shows how much he gets paid
depending on the number of items sold.
Use the table to answer the following questions.
And we're told that the data is exactly linear.
There should also be a period here.
So, because the data is exactly linear,
if we were to plot these ordered pairs
on the coordinate plane, as we see here,
notice how we can sketch a line
passing through all six points.
For the first part, we're asked to find
the vertical intercept,
which is the same as the y-intercept,
and then we're also asked to find the average rate of change
which, again, because our data is exactly linear,
it's constant, and it would be the slope of the line
that we just saw passing through all six points.
Looking below, we have notes for a linear equation,
in terms of x and y, and also using function notation.
Let's begin by determining the average rate of change,
which would be the change in the function value
divided by the change of the input,
or, in our case, the change in pay,
divided by the change in the number of items sold.
And, because our table is linear,
we can pick any two ordered pairs to find this quotient.
Let's go ahead and use these first two ordered pairs,
and therefore the average rate of change,
which, again, would also be the slope of the line
passing through these six points,
would be equal to the change in the pay,
which is 304.5
minus 268.9
divided by the change in the number of items sold,
which is 50 minus 10.
This comes out to 35.6 divided by 40,
which is 0.89.
So, the average rate of change is 0.89,
which tells us he makes 89 cents for every item he sells.
And now, to find the vertical intercept,
which would be the pay
when the number of items sold is zero,
we'll set up the equation: P equals m times n plus b.
So we'd have P, the pay, is equal to the slope
times the input variable, or 0.89,
times n plus the unknown, b.
And now we'll pick one ordered pair, perform substitution
for n and P, and then solve for b.
Again, it doesn't matter which ordered pair we use,
since this data is linear.
Let's go and use this first ordered pair.
So we'll substitute 268.9 for P and 10 for n.
So this'll give us 268.9 equals 0.89 times 10 plus b,
so we have 268.9 equals 8.9 plus b.
We subtract 8.9 from both sides,
notice how b is equal to 260,
which means the vertical intercept
would be the ordered pair: zero comma 260.
This tells us, if Gustavo sells zero items,
he still gets paid 260 dollars,
which would be his base salary.
By finding the average rate of change
and the vertical intercept,
we also found the equation that relates P and n.
We now know that P is equal to 0.89n plus 260,
and we'll need this for the next part.
The next part asks us to use the results from part a
to write the linear function that represents the data.
So, we just found this equation here,
but to use function notation, we replace P with P of n,
again, where n is the input variable.
The linear function would be P of n equals
0.89n plus 260.
Next, we're asked to use the function
to determine how much Gustavo will be paid
if he sells 440 lightsabers,
which means you want to find P of 440,
so we'll substitute 440 for n.
So, P of 440 is equal to 0.89 times 440 plus 260.
This'd be 391.6 plus 260,
which comes out to 651.6.
So, P of 440 equals 651.6.
As an ordered pair, this'd be
440 comma 651.6.
The meaning of this is that,
if Gustavo sells 440 lightsabers,
he will earn 651 dollars and 60 cents.
Notice how here the dollar sign is already included for us.
Now, for the last question, we're asked to use our function
to determine the number of lightsabers sold
if Gustavo's pay is 811 dollars and 80 cents,
which means P of n is equal to 811.8.
We want to find n.
To do this, we would have the equation:
811.8 equals 0.89n plus 260.
So, we'll subtract 260 on both sides.
This gives us 551.8 equals 0.89n.
Divide both sides by 0.89.
This gives us n equals 620.
So, n equals 620.
As an ordered pair, remember the input is first,
so we have 620 comma the function value or output of 811.8,
which means, Gustavo sold 620 lightsabers
to earn 811 dollars and 80 cents.
I hope you found this helpful.