Question 1209648: Determine the range of the function $(x) = -4x^2-8x-6 + 3x^2 - 17x + 10. Enter your answer in interval notation.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 1. **Simplify the function:**
Combine like terms:
f(x) = (-4x² + 3x²) + (-8x - 17x) + (-6 + 10)
f(x) = -x² - 25x + 4
2. **Find the vertex:**
The x-coordinate of the vertex is given by x = -b / 2a, where a = -1 and b = -25.
x = -(-25) / (2 * -1) = 25 / -2 = -12.5
3. **Find the y-coordinate of the vertex:**
Substitute x = -12.5 into the simplified function:
f(-12.5) = -(-12.5)² - 25(-12.5) + 4
f(-12.5) = -156.25 + 312.5 + 4
f(-12.5) = 160.25
4. **Determine the range:**
Since the coefficient of the x² term is negative, the parabola opens downward. This means the vertex represents the maximum value of the function. The range will be all y-values less than or equal to the y-coordinate of the vertex.
Therefore, the range of the function is $(-\infty, 160.25]$.
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