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

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

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

Answer

Explanation:

Step1: Identify the function type

The function ( f(x)=-x^{2}+3x + 8 ) is a quadratic function in the form ( y = ax^{2}+bx + c ), where ( a=-1), (b = 3), (c = 8).

Step2: Find the axis of symmetry

The formula for the axis of symmetry of a quadratic function ( y=ax^{2}+bx + c ) is ( x=-\frac{b}{2a} ). Substitute ( a=-1) and ( b = 3 ) into the formula: ( x=-\frac{3}{2\times(-1)}=\frac{3}{2}=1.5 )

Step3: Determine the increasing interval

For a quadratic function ( y = ax^{2}+bx + c ), if ( a<0 ), the parabola opens downwards. Since ( a=-1<0 ), the function ( f(x)) is increasing to the left of the axis of symmetry.

Answer:

((-\infty,1.5))