Question 1209637: For parts (a) through (d), let f(x) = 2x^2 - 13x + 20 - 5x^2 + 19x + 7.
What is the vertex of the parabola y = f(x)?
Enter your answer as a coordinate pair, that is, as $(x,y)$ where $x$ and $y$ are replaced by numbers.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the vertex of the parabola:
1. **Simplify the function:**
Combine like terms in the given function:
f(x) = (2x² - 5x²) + (-13x + 19x) + (20 + 7)
f(x) = -3x² + 6x + 27
2. **Find the x-coordinate of the vertex:**
The x-coordinate of the vertex of a parabola in the form ax² + bx + c is given by:
x = -b / 2a
In our simplified equation, a = -3 and b = 6. So:
x = -6 / (2 * -3)
x = -6 / -6
x = 1
3. **Find the y-coordinate of the vertex:**
Substitute the x-coordinate (x = 1) back into the simplified function to find the y-coordinate:
f(1) = -3(1)² + 6(1) + 27
f(1) = -3 + 6 + 27
f(1) = 30
4. **Vertex coordinates:**
The vertex of the parabola is (x, y) = (1, 30).
Therefore, the vertex of the parabola y = f(x) is $(1, 30)$.
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