Find the intervals on which​ f(x) is increasing and the intervals on which​ f(x) is decreasing.  Then sketch the graph.  Add horizontal tangent lines. f(x)=4+8x-x^2

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 \( (4, 20) \).

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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.

Find the intervals on which f(x) is increasing and the intervals on which f(x) is decreasing. Then sketch the graph. Add horizontal tangent lines. f(x) = 4 + 8x - x^2

Discover how to find the intervals where the quadratic function f(x) = 4 + 8x - x^2 is increasing or decreasing. This solution includes a step-by-step guide, use of the first derivative test, and a sketch of the graph with horizontal tangent lines. Suitable for high school students studying calculus or advanced algebra.

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