Question

PLEASE PLEASE HELP! Algebra!!!

Answer 1

The graph below shows a company's profit f(x), in dollars, depending on the price of erasers x, in dollars, being sold by the company:

Answer 2

Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit? (6 points)

Part B: What is an approximate average rate of change of the graph from x = 1 to x = 4, and what does this rate represent? (4 points)

Answer 3

@ganeshie8 @pooja195 @satellite73 @Loser66 @UsukiDoll

Answer 4

@mrdoldum @NeonStrawsForever @sweetburger @freckles

Answer 5

@mafhater Don't have time now, but if you can wait a few hours I'll help you through it.

Answer 6

So, let's tackle this a little more methodically.
F(x) is the profit of the company depending on "x" the price of the erasers being sold.
Now, we can observe a parabola whose vertex is defined by the point (4,270).
But what does that mean?: It means that the profit is "maximum" when the erasers have the price of 4 dollars, how do I know? because there is no point whose image (f(x)) is higher than the point (4,270).

The x-intercepts are the prices of the erasers that generate no profit or very low at least, and the values below the x-axis mean that there has been loss in profit.
So, we can conclude that:
\[f(0)=0\]
\[f(8)=0\]
This two expressions of functions mean that the profit F(x) is 0 when the erasers have the price of 0 and 8 dollars.

In order to know when a function is increasing or decreasing, by definition it is notated like:
\[f \rightarrow inc(a) <=> \forall x \in E(a,(a +\delta));f(a)<f(x)\]
\[f \rightarrow decr.(a)<=> \forall x \in E((a- \delta),a) ; f(a)>f(x) \]
This is notated in mathematics notation, but I'll translate it as simple as I can:
"the function 'f' is increasing on the point 'a' only if all the x values belonging in the interval a to a+delta have images greater than the image of the point 'a'"
"The function 'f' is decreasing on the point 'a' only if all the x values belongin to the interval 'a - delta" to 'a' have images minor to the image of the point 'a'"

Having this in minf, we have to look the functions graph from lefto to right and look at the points, we can see that the images are getting higher as we move to the right of the origin to the positive values of x so we can say the function is increasing as we move towards the maximum point we established, which is "(4,270)".
And as we move further to the right of the point "(4,270)" we can see that the images are getting lower, meaning that the function is decreasing.
These intervals can be written as:
\[( -\inf,4) \cup (4, + \inf)\]
Note that the parenthesis are rounf because of two reasons:
(1) The infinity always have round parenthesis when notating the interval
(2) since x=4 is the maximum point of the function, we do not include it as decreasing or increasing because it is an "absolute maximum point" since no other x value in the domain has greater image than x=4.

This is how I'd explain the part A.

Answer 7

Wow!! impressive explanation!!

Answer 8

For part "b", we will refer to the "rate of change" formula or for algebra also known as the "slope" formula:
\[m=\frac{ f(b)-f(a) }{ b-a }\]
The conditions for this is that (b-a) must not be equal to zero, because that would give an indetermined rate of change (some books call it infinite) and you can never have an infinite rate of change on a curve in this case.
So, with this, we can calculate the rate of change between two points, which in this case are (1,60) and (4,270) (the maximum point).
So, we will take "a" as the point (1,60), and "b" as the point (4,270) so therefore, f(a) is 60 and f(b) is 270, and "a" is equal to 1 and "b" is equal to 4, because they are values of "x" we choose.
So, let's now apply the slope formula for this two points:
\[m=\frac{ 270-60 }{ 4-1 }\]
And do the corresponding operations:
\[m=\frac{ 210 }{ 3 }\]
\[m=70\]
We can now conclude that the average rate of change is "70" but; What is this number?.
Well, this "70" in this specific function where x is the "price" and f(x) is the "profit", is called the "Marginal profit", and it represents the variation of the profit as we change the price of the unit.

Now, let's focus on the sign for a moment, and to make it more evident: We can observe that the m=70 does not have a "-" sign before it, so we can say that it is positive, so we can reprsent it as: m=+70.
This means that as we changes the price of the erasers from 1$ each to 4$ each we have an increase of profit, meaning that it is convenient for the Company to increase the price of the erasers to 4$.