Question

over which interval is the graph of ( f(x) = -x^2 + 3x + 8 ) increasing?
( \bigcirc (-infty, 1.5) )
( \bigcirc (-infty, 10.25) )
( \bigcirc (1.5, infty) )
( \bigcirc (10.25, infty) )

Explanation:

Step1: Identify the parabola's direction

The function is \( f(x) = -x^2 + 3x + 8 \). Since the coefficient of \( x^2 \) is -1 (negative), the parabola opens downward.

Step2: Find the vertex's x - coordinate

For a quadratic function \( f(x)=ax^{2}+bx + c \), the x - coordinate of the vertex is given by \( x =-\frac{b}{2a} \). Here, \( a=- 1 \) and \( b = 3 \). So \( x=-\frac{3}{2\times(-1)}=\frac{3}{2}=1.5 \).

Step3: Determine the increasing interval

For a downward - opening parabola, the function is increasing to the left of the vertex (before the x - coordinate of the vertex) and decreasing to the right of the vertex (after the x - coordinate of the vertex). So the function \( f(x)=-x^{2}+3x + 8 \) is increasing on the interval \( (-\infty,1.5) \).

Answer:

\( (-\infty, 1.5) \) (Option: \( (-\infty, 1.5) \))