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 \( f(x) = -x^2 + 3x + 8 \). The coefficient of \( x^2 \) is \(-1\), which is negative. So 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

Since the parabola opens downward, the function is increasing to the left of the vertex (before the vertex) and decreasing to the right of the vertex (after the vertex). So the function is increasing on the interval \( (-\infty, 1.5) \).

Answer:

\( (-\infty, 1.5) \) (corresponding to the first option: \( (-\infty, 1.5) \))