Over which interval is the graph of f(x) = –x2 + 3x + 8 increasing?

Hello there!

To find the increasing intervals for this graph just based on the equation, we should find the turning points first.

Take the derivative of f(x)...
f(x)=-x²+3x+8
f'(x)=-2x+3

Set f'(x) equal to 0...
0=-2x+3
-3=-2x
3/2=x

This means that the x-value of our turning point is 3/2. Now we need to analyze the equation to figure out the end behavior of this graph as x approaches infinity and negative infinity.
Since the leading coefficient is -1, as x approaches ∞, f(x) approaches -∞ Because the exponent of the leading term is even, the end behavior of f(x) as x approaches -∞ is also -∞.

This means that the interval by which this parabola is increasing is...
(-∞,3/2)

PLEASE DON'T include 3/2 on the increasing interval because it's a turning point. The slope of the tangent line to the turning point is 0 so the graph isn't increasing OR decreasing at this point.

I really hope this helps!
Best wishes :)