1. The problem is to understand where a function decreases on its graph. 2. A function decreases on intervals where its derivative is less than zero, i.e., $f'(x) < 0$. 3. To find these intervals, first find the derivative $f'(x)$ of the function $f(x)$. 4. Solve the inequality $f'(x) < 0$ to find the intervals where the function decreases. 5. For example, if $f(x) = -x^2 + 4x + 1$, then $f'(x) = -2x + 4$. 6. Solve $-2x + 4 < 0$: $$ -2x + 4 < 0 $$ 7. Subtract 4 from both sides: $$ -2x + \cancel{4} - \cancel{4} < 0 - 4 $$ $$ -2x < -4 $$ 8. Divide both sides by $-2$ and reverse inequality sign because dividing by a negative number: $$ \frac{-2x}{\cancel{-2}} > \frac{-4}{\cancel{-2}} $$ $$ x > 2 $$ 9. So, the function decreases for $x > 2$. 10. This means the graph goes downwards when $x$ is greater than 2. This is how you find where a function decreases by using its derivative.