Math 151
Summer 2019
Summer 2019
Dear Aunt Sally –
I am
glad to hear that your garden is growing so well and that the dog hasn’t given
you any trouble. As you know, I am
taking a College Algebra course this summer.
I am learning all kinds of interesting things about mathematical
relationships. We are specifically focused on learning different types of
functions. A function is a kind of
relationship, where one thing equals only one other value.
It’s like when you go shopping
sales. Let’s say you found Cheerios on
sale for $3.99 per box, the relationship between the box’s individual cost, how
many boxes you would buy, and your total cost would be a function. See, the number of boxes you would buy is known
as an independent variable, which we can represent as just the letter x. It is also an independent variable, or just
an input. We know that every box is
$3.99, which is a constant, because the price stays the same no matter how many
boxes of Cheerios you buy. The final
cost is your dependent variable, or output, (which can be written as f(x) or “f
of x” as we call it in class) because it depends on how many boxes you
buy.
This can be represented by the
equation f(x)=3.99x, where 3.99 is the cost per box, x is the number of boxes,
and f(x) or Y is the total cost. In my
Algebra class, we would have called this kind of a function a “one-to-one”
function, which means that each x value, or the number of boxes you buy, is
equal to only one cost. If you buy one box, it is $3.99. If you buy 2 boxes, it is $7.98 and so
on. You will not pay $7.98 for only one
box or for three or more boxes. You will
only ever pay $7.98 for two boxes. No
more, no less.
We’re also learning how to graph
various functions on a graph, and what that means. There are many ways that we can graph a
function, but the most common way to graph a line is by x- and
y-intercept. This is the point where the
line passes the x-axis (or y-axis) on a graph. Another way to say this is that
it is the value of x when y is zero, or vice versa. If you bought zero boxes of cheerios, you
would spend $0, so the line would include a point where zero equals zero or the
point (0,0) where the first 0 is the number of boxes (or x) and the second
value (the y value) is how much it cost total, which is also 0. Another point on the graph would be (2,
$7.98) and (3, $11.97) and so on. The
total x values, the values {0, 1, 2, 3…}, represent the “domain” of the
function. The domain is the set of all
of the input variables as a list. The
range is the set of all outputs, so {3.99, 7.98, 11.97…). With the cereal relationship, the graph
starts at 0 boxes for $0, then one box for $3.99, two for $7.98, three for
$11.97 and so on. This creates a
straight line up the graph at a 90-degree angle from the point of origin. We would call this a linear function because
it makes a straight line!
There are many kinds of
relationships in math, and especially as they relate to functions, but I fear
that I have already bored you to tears.
I look forward to your next correspondence!
Love,
Katlin <3
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Score: 10/10
Professor Comments: Hi (Katlin), great job on this rough draft. This is really a very good submission even for a final draft. There are a couple of things you could do to improve your letter. Maybe provide a picture of the cereal box graph that you describe. Or possibly, provide a second real-life example or even a non-example to contrast something that is not a function. Either of these I think would help to solidify the concept for aunt Sally.
Your writing mechanics are great and you clearly understand what is a function. You do make a reference to a one-to-one function which is not correct so revisit that statement. Overall, you have a really solid letter already.
I hope this helps you focus your final draft. I have added full credit for the rough draft (quiz grade). Good job!
