To analyze the quadratic function \( y = x^2 - 16x + 48 \), let's identify the key points: the vertex, x-intercepts, and y-intercept.
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Vertex: The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] For the equation \( y = x^2 - 16x + 48 \), we have \( a = 1 \) and \( b = -16 \). Plugging in these values: \[ x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \] To find the y-coordinate, substitute \( x = 8 \) back into the equation: \[ y = (8)^2 - 16(8) + 48 = 64 - 128 + 48 = -16 \] Thus, the vertex is \( (8, -16) \).
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Y-intercept: The y-intercept occurs where \( x = 0 \): \[ y = 0^2 - 16(0) + 48 = 48 \] Thus, the y-intercept is \( (0, 48) \).
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X-intercepts: The x-intercepts occur where \( y = 0 \). To find the x-intercepts, we solve the equation: \[ 0 = x^2 - 16x + 48 \] We can factor this quadratic: \[ 0 = (x - 12)(x - 4) \] Setting each factor to zero gives: \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] \[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \] Thus, the x-intercepts are \( (12, 0) \) and \( (4, 0) \).
Summary of Key Points:
- Vertex: \( (8, -16) \)
- Y-intercept: \( (0, 48) \)
- X-intercepts: \( (12, 0) \) and \( (4, 0) \)
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