what is the vertex and ra nge?

Question

what is the vertex and determine a max or minimun value and find that value. what is the range in inerval nation. using interval notation indicate when the function is increasing and decreasing.

-3x^2+24x-49

Answer 1

-3x^2+24x-49
dy/dx=-6x +24
-6x+24 =0, so x=4

So the max occurs when x=4. ( We know it is a max because the coefficient of x^2 is negative.)
so the max is -3(4^2) +2*4-49 = -48 +8 -49= -89
This means that the range (possible values of y) is all the real numbers =< -89 . Because -89 is the max value. In interval notation this is - infinity
The graph is a parabola which is increasing upward until it hits the max when x = 4 and then decreases forever
Thus f(x) is increasing over the interval - infinity
The vertex is located at (4, -89)

Since f(x) is never positive, it never crosses the x-axis and thus has no real roots. The two roots are imaginary.

I think that covers all your questions.

Answer 2

He/ she just made a small error down below, the max is -3(4)^2+(24*4)-49=-1 i think.