To find the vertex of the parabola given by the equation \( y = x^2 - 3x \), we can use the formula for the vertex of a parabola in the form \( y = ax^2 + bx + c \). The vertex \( (h, k) \) can be found with the following calculations:
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Identify the coefficients:
- \( a = 1 \)
- \( b = -3 \)
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Calculate the \( x \)-coordinate of the vertex using the formula: \[ h = -\frac{b}{2a} = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \]
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Substitute \( h \) back into the equation to find the \( y \)-coordinate \( k \): \[ k = \left( \frac{3}{2} \right)^2 - 3\left( \frac{3}{2} \right) \] \[ = \frac{9}{4} - \frac{9}{2} \] To subtract these fractions, convert \(\frac{9}{2}\) to have a common denominator of 4: \[ \frac{9}{2} = \frac{18}{4} \] Now, perform the subtraction: \[ k = \frac{9}{4} - \frac{18}{4} = \frac{9 - 18}{4} = \frac{-9}{4} \]
Thus, the vertex of the parabola \( y = x^2 - 3x \) is: \[ \left( \frac{3}{2}, -\frac{9}{4} \right) \]
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