The equation you provided, \( P = -5x^2 + 1000x + 5000 \), represents a quadratic function of the form \( P(x) = ax^2 + bx + c \), where:
- \( a = -5 \)
- \( b = 1000 \)
- \( c = 5000 \)
This is a downward-opening parabola since the coefficient of \( x^2 \) (which is \( a \)) is negative.
To analyze this function, we can find several important features, such as the vertex, intercepts, and the maximum value:
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Vertex: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} \]
Substituting in the values of \( a \) and \( b \):
\[ x = -\frac{1000}{2 \times -5} = -\frac{1000}{-10} = 100 \]
To find the corresponding \( P \) value at the vertex, plug \( x = 100 \) back into the equation:
\[ P(100) = -5(100)^2 + 1000(100) + 5000 \] \[ = -5(10000) + 100000 + 5000 \] \[ = -50000 + 100000 + 5000 = 50000 \]
Therefore, the vertex of the parabola is at \( (100, 50000) \), which is the maximum point since the parabola opens downwards.
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Intercepts:
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Y-intercept: This occurs when \( x = 0 \): \[ P(0) = -5(0)^2 + 1000(0) + 5000 = 5000 \] So the y-intercept is \( (0, 5000) \).
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X-intercepts: To find the x-intercepts, set \( P = 0 \): \[ -5x^2 + 1000x + 5000 = 0 \] We can divide all terms by -5: \[ x^2 - 200x - 1000 = 0 \]
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ x = \frac{200 \pm \sqrt{(-200)^2 - 4(1)(-1000)}}{2(1)} \] \[ = \frac{200 \pm \sqrt{40000 + 4000}}{2} \] \[ = \frac{200 \pm \sqrt{44000}}{2} \] \[ = \frac{200 \pm 20\sqrt{11}}{2} \] \[ = 100 \pm 10\sqrt{11} \]
Thus, the x-intercepts are: \[ x_1 = 100 + 10\sqrt{11}, \quad x_2 = 100 - 10\sqrt{11} \]
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In summary, the function \( P = -5x^2 + 1000x + 5000 \) has a maximum value of 50000 at \( x = 100 \), a y-intercept at \( (0, 5000) \), and two x-intercepts at \( x = 100 \pm 10\sqrt{11} \).