helpp im super bad at math
(the picture shown is a graph that increases by 40 on the y axis and 2 on the x axis. the parabola is an upward opening one that has a maximum of (5,160) and x intercepts of (0,0) and (10,0) the graph represents a company's profit f(x), in dollars, depending on the price of pencils x, in dollars, sold by the company)
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?
(i answered part of it):
the x intercepts shows that the company made no profit when the price of pencils were (0,0) and (0,10). the maximum shows that the company made the most profit at (5,160). (is it right?? if its not plss correct it)
Part B: What is an approximate average rate of change of the graph from x = 2 to x = 5, and what does this rate represent?
Part C: Describe the constraints of the domain.
a parabola that opens up has no maximum.
If the vertex is at (5,160) then the equation is
y = a(x-5)^2 + 160
Since (0,0) is on the graph, that means
a(0-5)^2 + 160 = 0
a = -160/25 = -32/5
So the parabola opens down, not up
y = -32/5 (x-5)^2 + 160
now see what you can do with A and B
ty the main thing i rlly need help on is the interval and the change of rate part
(i messed up on the description i meant downward opening (it wont let me paste the pic))
the avg rate of change is the slope of the line joining the two points on the curve
(2,f(2)) and (5,f(5))
it will be in units of $/pencil
Part A: Your answer is partially correct. The x-intercepts represent the points where the graph crosses the x-axis. In this case, the x-intercepts are (0,0) and (10,0), which means that the company did not make any profit when the price of pencils was $0 or $10. The maximum value of the graph, which is (5,160), represents the point where the graph reaches its highest value or peak. In this case, it means that the company made a profit of $160 when the price of pencils was $5.
Regarding the intervals where the function is increasing or decreasing, we can analyze the shape of the graph. Since the graph is a parabola that opens upwards, it means that the function is increasing as you move from left to right until it reaches the point of maximum profit. Beyond the maximum point, the function starts decreasing again.
Part B: To calculate the approximate average rate of change from x = 2 to x = 5, we need to find the change in y-values (profit) divided by the change in x-values (price of pencils).
First, let's find the y-values corresponding to x = 2 and x = 5. By looking at the graph, we can estimate that the y-values are around 100 and 140, respectively.
Change in y-values = 140 - 100 = 40
Change in x-values = 5 - 2 = 3
Average rate of change = (Change in y-values) / (Change in x-values) = 40 / 3
The approximate average rate of change for the given interval is approximately 13.33. This rate represents the average increase in profit per dollar increase in the price of pencils within the given interval.
Part C: The constraints of the domain refer to any limitations or restrictions on the values that x (price of pencils) can have. In this case, the graph and problem statement do not provide any specific constraints on x. However, commonly seen constraints in real-world scenarios could be price limitations (e.g., x cannot be negative) or practical limitations (e.g., x must be within a certain range). Without additional information, we can assume that the domain of x is all real numbers from negative infinity to positive infinity.