To determine the domain and range of the quadratic function \( y = -x^2 + 2x - 3 \), we start by identifying its properties.
Domain:
The function is a polynomial, specifically a quadratic function, which is defined for all real numbers. Therefore, the domain is: \[ \text{Domain: } (-\infty, \infty) \]
Range:
To find the range, we need to determine the vertex of the parabola represented by the function. The function is in standard form \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = 2 \), and \( c = -3 \).
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Find the x-coordinate of the vertex using the formula \( x = -\frac{b}{2a} \): \[ x = -\frac{2}{2 \cdot -1} = 1 \]
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Substitute \( x = 1 \) back into the equation to find the y-coordinate of the vertex: \[ y = - (1)^2 + 2(1) - 3 = -1 + 2 - 3 = -2 \]
The vertex of the parabola is at the point \( (1, -2) \). Since the coefficient of \( x^2 \) (which is \( a = -1 \)) is negative, the parabola opens downward.
The maximum value of \( y \) occurs at the vertex, and the function approaches negative infinity as \( x \) moves away from the vertex. Therefore, the range is: \[ \text{Range: } (-\infty, -2] \]
Summary:
- Domain: \( (-\infty, \infty) \)
- Range: \( (-\infty, -2] \)