To analyze the quadratic function \( y = x^2 - 2x - 120 \), we can determine the vertex, x-intercepts, and y-intercept.
1. Vertex
The vertex of a quadratic function in the standard form \( ax^2 + bx + c \) can be calculated using the formula: \[ x = -\frac{b}{2a} \] For \( y = x^2 - 2x - 120 \):
- \( a = 1 \)
- \( b = -2 \)
Calculating the x-coordinate of the vertex: \[ x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1 \]
Substituting \( x = 1 \) back into the equation to find the y-coordinate: \[ y = (1)^2 - 2(1) - 120 = 1 - 2 - 120 = -121 \] Thus, the vertex is \( (1, -121) \).
2. x-intercepts
To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 2x - 120 \] Factoring the quadratic: \[ 0 = (x - 12)(x + 10) \] Setting each factor to zero gives us the x-intercepts:
- \( x - 12 = 0 \) → \( x = 12 \)
- \( x + 10 = 0 \) → \( x = -10 \)
Therefore, the x-intercepts are \( (-10, 0) \) and \( (12, 0) \).
3. y-intercept
To find the y-intercept, set \( x = 0 \): \[ y = (0)^2 - 2(0) - 120 = -120 \] Thus, the y-intercept is \( (0, -120) \).
Summary of Key Points:
- Vertex: \( (1, -121) \)
- x-intercepts: \( (-10, 0) \) and \( (12, 0) \)
- y-intercept: \( (0, -120) \)
Final Response
- Vertex: \( (1, -121) \)
- x-intercepts: \( (-10, 0) \) and \( (12, 0) \)
- y-intercept: \( (0, -120) \)