Question

It's also useful to see this behavior represented graphically. Each time we increase x by 1 unit, the change in y increases by 2 , including increasing from negatrve yalues for Δy to positive values for Δy. Each time x increases by 1 , the change in y (the value of Δy ) increases by 2 (from -5 , to -3 , to -1 , to 1 , to 3 , to 5 as shown here). Even with this expanded domain, that the change in the function values follow a specific pattern. The change in Δy itself changes by a constant amount (the y-value always changes by 2 units more than the previous change when x increases by 1 ) This is a key aspect of what it means for a function to be quadratic. Score on last attempt: 0.5 out of 3 (parts; 0.0.5,$0.5,0.5,∗0.0.5,260.0.5,300.5,40.0.5 ) Use the function f(x)=x² to answer the following questions. (Its table of vatues and grapi are given above) a. The change in y from x=−3 to x=−2 is b. The change in y from x=1 to x=2 is c. The average rate of change from x=−3 to x=−1 is d. The ayerage rate of change from x=1 to x=3 is e. This fianction ha the average rates of change ov er equally-sized intervals always decrease as x increases. the average rates of change over equally-sized intervals always increase as x increases. the average rates of change over equally-sized intervals somethes increase and sometimes decrease as a increases.

Answer 1

a. The change in y from x=-3 to x=-2 is 5. b. The change in y from x=1 to x=2 is 3. c. The average rate of change from x=-3 to x=-1 is -4. d. The average rate of change from x=1 to x=3 is 4. e. The function has the average rates of change over equally-sized intervals sometimes increase and sometimes decrease as x increases.

Step-by-step explanation:

a. The change in y from x=-3 to x=-2 is:
To find the change in y, we need to substitute the values of x into the function f(x)=x². So, when x=-3, y=(-3)² = 9. And when x=-2, y=(-2)² = 4. Therefore, the change in y from x=-3 to x=-2 is 9-4=5.

b. The change in y from x=1 to x=2 is:
Using the same process, we find that when x=1, y=1² = 1. And when x=2, y=2² = 4. So, the change in y from x=1 to x=2 is 4-1=3.

c. The average rate of change from x=-3 to x=-1 is:
The average rate of change can be found by subtracting the initial value of y from the final value of y and then dividing by the change in x. So, when x=-3, y=(-3)² = 9. And when x=-1, y=(-1)² = 1. The change in x is -3-(-1) = -2. Therefore, the average rate of change from x=-3 to x=-1 is (1-9)/(-2) = -4.

d. The average rate of change from x=1 to x=3 is:
Following the same steps, when x=1, y=1² = 1. And when x=3, y=3² = 9. The change in x is 3-1 = 2. So, the average rate of change from x=1 to x=3 is (9-1)/2 = 4.

e. This function has the average rates of change over equally-sized intervals sometimes increase and sometimes decrease as x increases.
From the previous calculations, we can see that the average rate of change is not consistently increasing or decreasing as x increases. Therefore, the correct statement is that the average rates of change over equally-sized intervals sometimes increase and sometimes decrease as x increases.