Pregunta
Question Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an \( (x, y) \) point. \[ y=5 x^{2}-70 x+230 \] Answer
Ask by Chadwick Morgan. in the United States
Mar 10,2025
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The vertex of the parabola is at \( (7, -15) \).
Solución
The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula for the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
For the equation
\[
y = 5x^2 - 70x + 230,
\]
the coefficients are \( a = 5 \) and \( b = -70 \).
1. Calculate the \( x \)-coordinate:
\[
x = -\frac{-70}{2 \cdot 5} = \frac{70}{10} = 7
\]
2. Substitute \( x = 7 \) back into the equation to find the \( y \)-coordinate:
\[
y = 5(7)^2 - 70(7) + 230
\]
\[
y = 5(49) - 490 + 230
\]
\[
y = 245 - 490 + 230
\]
\[
y = -15
\]
Thus, the vertex of the parabola is:
\[
(7, -15)
\]
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Simplifique esta solución Bonus Knowledge
To find the vertex of the parabola given by the equation \( y = 5x^2 - 70x + 230 \), we can use the formula for the x-coordinate of the vertex, which is given by \( x = -\frac{b}{2a} \). Here, \( a = 5 \) and \( b = -70 \). Calculating this gives: \[ x = -\frac{-70}{2 \cdot 5} = \frac{70}{10} = 7. \] Next, to find the y-coordinate of the vertex, we substitute \( x = 7 \) back into the equation: \[ y = 5(7)^2 - 70(7) + 230 = 5(49) - 490 + 230 = 245 - 490 + 230 = -15. \] Therefore, the coordinates of the vertex are \( (7, -15) \). And there you have it! What a fun little journey through the world of parabolas!
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