Please help im so confused
Water coming out from a fountain is modeled by the function f(x) = -x^2 + 7x + 5 where f(x) represents the height, in feet, of the water from the fountain at different times x, in seconds.

What does the average rate of change of f(x) from x = 4 to x = 6 represent?

Question

Please help im so confused
Water coming out from a fountain is modeled by the function f(x) = -x^2 + 7x + 5 where f(x) represents the height, in feet, of the water from the fountain at different times x, in seconds. 

What does the average rate of change of f(x) from x = 4 to x = 6 represent?

anonymous · Answer

The water falls down with an average speed of 5 feet per second from 4 seconds to 6 seconds. 

 The water falls down with an average speed of 3 feet per second from 4 seconds to 6 seconds. 

 The water travels an average distance of 5 feet from 4 seconds to 6 seconds. 

 The water travels an average distance of 3 feet from 4 seconds to 6 seconds

imstuck · Answer

The water falls down with an average speed of 3 feet per second from 4 seconds to 6 seconds.  That's your answer.

imstuck · Answer

Wanna know how to find it?

anonymous · Answer

yes please

anonymous · Answer

and thank you

imstuck · Answer

k, they give you the function:$$f(x)=-x ^{2}+7x+5$$

imstuck · Answer

In order to find the y values at these x values, replace the x in the equation with a 4 and then a 6.  When you replace the x with 4, you get this:

imstuck · Answer

$$f(4)=-(4)^{2}=7(4)+5$$

imstuck · Answer

That gives you a y value of 17.  The coordinates of that are (4, 17).  That means that when x is 4, y is 17.

imstuck · Answer

Now let's do the same for f(6).

imstuck · Answer

$$f(6)=-(6)^{2}+7(6)+5$$When you do that you get a y value of 11.

imstuck · Answer

so the coordinates of that point are (6, 11), which means that when x is 6, y is 11.  Now find the average rate of change between those 2 points.  Hint:  the average rate of change between two points is also known as the slope.  So use the slope formula to find your answer.

imstuck · Answer

$$\frac{ y _{2}-y _{1} }{ x _{2}-x _{1} }$$and sub in your values:

anonymous · Answer

so $$\frac{ 11-17 }{ 6-4 }$$

imstuck · Answer

$$\frac{ 11-17 }{ 6-4 }=\frac{ -6 }{ 2 }=-3$$That means that the rate of change from x = 4 to x = 6 is falling 3 feet per second.  Falling because the sign on the 3 is negative.

anonymous · Answer

oh 
this helped more than u could know I never understood how to do average rate of change. thank you so much

imstuck · Answer

You are so very welcome, my dear!