math problem in calculas

Question

y=-9x^2-2x-7

i need to figure out where the interval increases and decreases.

Answer 1

y is represented by an inverted parabola. All such parabolas are increasing towards the vertex (maximum point) and decreasing after reaching the vertex. So we need to find the vertex.

y=-(9x²+2x+7)=-9(x²+2x/9+7/9)=-9(x²+2x/9+1/81-1/81+7/9),

y=-9(x+1/9)²+9/81-7=-9(x+1/9)²-62/9. When x=-1/9, y=-62/9, its maximum value, because subtracting 9(x+1/9)² from this maximum always decreases the value. Therefore, (-1/9,-62/9) is the vertex, and when -∞<x<-1/9 (interval (-∞,-1/9)), y is increasing and -1/9<x<∞ (interval (-1/9,∞)) y is decreasing.