Here is the graph of the function $f(x) = 4 + 8x - x^2$. The red point marks the vertex at $(4, 20)$, and the green dashed line represents the horizontal tangent at this maximum point.

Summary:

• The function is increasing on the interval $(-\infty, 4)$.
• The function is decreasing on the interval $(4, \infty)$.
• There is a local maximum at $(4, 20)$.

Would you like further details or have any questions?

Here are five related questions:

1. How can you determine concavity from the second derivative?
2. What is the significance of horizontal tangent lines in a graph?
3. How does changing the coefficient of $x^2$ affect the shape of the parabola?
4. What are the x-intercepts of the function $f(x) = 4 + 8x - x^2$?
5. Can you find the area between this parabola and the x-axis?

Tip: When analyzing quadratic functions, always consider the vertex, intercepts, and direction of opening to sketch a reliable graph.