An equation is y = The average rate of change is

Question

Write an equation of the parabola that passes through the point (62,−490)​ and has x-intercepts -8 and 72. Then find the average rate of change from x=-8 to x=2​ 

Answer 1

Write an equation of the parabola that passes through the point (62,−490)​ and has x-intercepts -8 and 72. Then find the average rate of change from x=-8 to x=2​

A concave-up or concave-down parabola looks like (x-h)²=4p(y-k) where (h,k) is the vertex and p=the directrix (the horizontal line y=p).

A concave-left or concave-right parabola looks like (y-h)²=4p(x-k) where (h,k) is the vertex and p=the directrix (the vertical line x=p).

Since this one intercepts the x-axis in two places, it must be concave-vertical.

We know three points on the parabola: (62,-490), (-8,0), and (72,0)

(x-h)²=4p(y-k)

If you plug in those three coordinates you'll get three equations in the three unknowns h,k,p. I'll let you do that.

Once you get the constants h, k, and p, then

(d/dt)[(x-h)²]=(d/dt)[4p(y-k)]=4p(d/dt)(y-k)

2(x-h)(dx/dt)=4p(dy/dt)

dy/dt=[(x-h)](dx/dt)/(2p)

and I'll let you take it from there.

If you want help schedule time with me.

Answer 2

First determine A so that A(x+8)(x-72) = -490 when x = 62

Then notice that the parabola will = 0 when x = 8.

The average rate of change then will be (1/10)*the value of the parabola at x=2.