To determine the interval over which the graph of the function [tex]\( f(x) = -x^2 + 3x + 8 \)[/tex] is increasing, follow these steps:
1. Find the derivative of the function [tex]\( f(x) \)[/tex]:
The first step is to differentiate the function [tex]\( f(x) \)[/tex] to find its derivative [tex]\( f'(x) \)[/tex]. The derivative of the function gives us the slope of the tangent line at any point [tex]\( x \)[/tex] on the graph. Calculating the derivative,
[tex]\[
f'(x) = \frac{d}{dx}(-x^2 + 3x + 8) = -2x + 3
\][/tex]
2. Determine where the derivative is positive:
A function is increasing where its derivative is positive. So, we need to find the values of [tex]\( x \)[/tex] for which [tex]\( f'(x) > 0 \)[/tex].
Set the inequality:
[tex]\[
-2x + 3 > 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
-2x + 3 > 0
\][/tex]
[tex]\[
3 > 2x
\][/tex]
[tex]\[
\frac{3}{2} > x
\][/tex]
[tex]\[
x < 1.5
\][/tex]
Thus, [tex]\( f(x) \)[/tex] is increasing for [tex]\( x < 1.5 \)[/tex].
3. Write the interval where the function is increasing:
Based on the solution to the inequality, [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\( (-\infty, 1.5) \)[/tex].
Therefore, the interval over which the graph of [tex]\( f(x) = -x^2 + 3x + 8 \)[/tex] is increasing is:
[tex]\[
\boxed{(-\infty, 1.5)}
\][/tex]
