The x y-coordinate plane is given. The curve labeled f(x) = 0.05(8x − x2) enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at the origin, passes through the approximate point (1, 0.4), changes direction at the point (4, 0.8), goes down and right becoming more steep, and exits the window in the first quadrant.

The curve you're describing is given by the function \( f(x) = 0.05(8x - x^2) \). Let's analyze it step by step.

### 1. Function Analysis:
The function can be written as:
\[
f(x) = 0.05(8x - x^2)
\]
This is a quadratic function, and it can be rewritten as:
\[
f(x) = 0.05(-x^2 + 8x)
\]
or equivalently:
\[
f(x) = -0.05x^2 + 0.4x
\]
This equation represents a downward-opening parabola.

### 2. Key Features of the Curve:
- **Vertex**: The vertex form of a parabola is given by \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. For this standard form, the vertex can be found using the formula \( h = -\frac{b}{2a} \).
  - Here, \( a = -0.05 \) and \( b = 0.4 \).
  - The x-coordinate of the vertex is \( x = -\frac{0.4}{2(-0.05)} = 4 \).
  - Substituting \( x = 4 \) into the function to find the y-coordinate:
  \[
  f(4) = -0.05(4^2) + 0.4(4) = -0.05(16) + 1.6 = -0.8 + 1.6 = 0.8
  \]
  - So, the vertex is at \( (4, 0.8) \).

- **Intercepts**:
  - **X-intercepts**: The function intersects the x-axis where \( f(x) = 0 \). Setting \( f(x) = 0 \):
  \[
  0 = -0.05x^2 + 0.4x
  \]
  Factor out \( x \):
  \[
  x(-0.05x + 0.4) = 0
  \]
  So, \( x = 0 \) or \( -0.05x + 0.4 = 0 \). Solving for \( x \) in the second factor:
  \[
  -0.05x = -0.4 \implies x = 8
  \]
  The x-intercepts are at \( x = 0 \) and \( x = 8 \).

  - **Y-intercept**: The y-intercept occurs where \( x = 0 \). From the equation:
  \[
  f(0) = 0
  \]
  So, the y-intercept is at the origin \( (0, 0) \).

### 3. Curve Behavior:
- **Third Quadrant**: The curve enters from the third quadrant, which means for some negative values of \( x \), \( f(x) \) is also negative. This aligns with the parabola's nature as it approaches negative infinity.
- **Crossing the Origin**: As noted, the curve crosses the origin \( (0, 0) \).
- **Direction Change**: The curve reaches its maximum at the vertex \( (4, 0.8) \), after which it changes direction.
- **Exiting in the First Quadrant**: After the vertex, the curve continues downwards, becoming steeper and passing through the first quadrant.

### Conclusion:
The curve behaves as described: entering from the third quadrant, becoming less steep as it approaches the vertex, crossing the x-axis at \( x = 0 \), reaching the maximum at \( x = 4 \), and then steeply decreasing as it exits in the first quadrant.

Would you like more details on any part of this analysis, or have any questions?

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Here are 5 related questions you might find interesting:
1. How would the graph change if the coefficient of \( x^2 \) was positive?
2. What are the implications of changing the constant \( 0.05 \) to a larger or smaller value?
3. How would the curve look if the vertex was at a different point?
4. What is the significance of the x-intercepts in relation to the graph's shape?
5. How would shifting the graph vertically or horizontally affect its position in the coordinate plane?

**Tip:** Always check the vertex and intercepts first when analyzing the shape and direction of a quadratic function's graph.

Explore the properties of the quadratic function f(x) = 0.05(8x - x^2), including its vertex, intercepts, and directional changes. This analysis covers key concepts in quadratic functions and is suitable for advanced high school students.

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