To analyze the quadratic function \(y = x^2 - 2x - 120\), we will find the x-intercepts, y-intercept, and vertex of the graph.
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Y-Intercept: The y-intercept occurs when \(x = 0\): \[ y = 0^2 - 2(0) - 120 = -120 \] Thus, the y-intercept is: \[ (0, -120) \]
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X-Intercepts: To find the x-intercepts, we set \(y = 0\) and solve for \(x\): \[ 0 = x^2 - 2x - 120 \] We can factor this quadratic equation: \[ 0 = (x - 12)(x + 10) \] Setting each factor to zero gives us: \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] \[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \] Therefore, the x-intercepts are: \[ (-10, 0) \quad \text{and} \quad (12, 0) \]
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Vertex: The vertex of a quadratic in the standard form \(y = ax^2 + bx + c\) can be found using the formula: \[ x = -\frac{b}{2a} \] For our function, \(a = 1\) and \(b = -2\): \[ x = -\frac{-2}{2 \cdot 1} = 1 \] To find the corresponding \(y\)-value, we substitute \(x = 1\) back into the equation: \[ y = 1^2 - 2(1) - 120 = 1 - 2 - 120 = -121 \] Thus, the vertex is: \[ (1, -121) \]
Finally, we can summarize the key points:
- x-intercepts: \((-10, 0)\) and \( (12, 0)\)
- y-intercept: \( (0, -120) \)
- vertex: \( (1, -121) \)
So:
- x-intercepts: \((-10, 0)\) and \( (12, 0)\)
- y-intercept: \( (0, -120) \)
- vertex: \( (1, -121) \)