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What are the key points on the graph of y=x^2−2x−120

? Name the vertex, x-intercept(s), and y-intercept.(1 point)
Responses

x-intercepts: (−10,0)
(12,0)

y-intercept: (1,−121)

vertex: (0,−120)

x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y-intercept: left parenthesis 1 comma negative 121 right parenthesis vertex: left parenthesis 0 comma negative 120 right parenthesis

x-intercepts: (−10,0)
(12,0)

y-intercept: (0,−120)

vertex: (1,−121)

x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis 1 comma negative 121 right parenthesis

x-intercept: (1,−121)

y-intercept: (0,−120)

vertex: (−10,0)
(12,0)

x -intercept: left parenthesis 1 comma negative 121 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis

x-intercept: (0,−120)

y-intercept: (−10,0)
(12,0)

vertex: (1,−121)

1 answer

To find the key points of the quadratic function \( y = x^2 - 2x - 120 \), we will identify the vertex, x-intercepts, and y-intercept.

  1. Vertex: The vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] Here, \( a = 1 \) and \( b = -2 \): \[ x = -\frac{-2}{2 \cdot 1} = 1 \] Plug \( x = 1 \) back into the equation to find the y-coordinate: \[ y = 1^2 - 2 \cdot 1 - 120 = 1 - 2 - 120 = -121 \] So, the vertex is \( (1, -121) \).

  2. Y-Intercept: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 2 \cdot 0 - 120 = -120 \] Therefore, the y-intercept is \( (0, -120) \).

  3. X-Intercepts: The x-intercepts occur when \( y = 0 \): \[ 0 = x^2 - 2x - 120 \] To find the roots of the quadratic equation, use factoring, completing the square, or the quadratic formula. Factoring gives: \[ 0 = (x - 12)(x + 10) \] Setting each factor to zero gives us the x-intercepts: \[ x - 12 = 0 \implies x = 12 \] \[ x + 10 = 0 \implies x = -10 \] Thus, the x-intercepts are \( (-10, 0) \) and \( (12, 0) \).

Summary

  • Vertex: \( (1, -121) \)
  • X-Intercepts: \( (-10, 0) \) and \( (12, 0) \)
  • Y-Intercept: \( (0, -120) \)

Therefore, the correct response, summarizing the findings, is:

  • x-intercepts: \( (-10, 0) \) and \( (12, 0) \)
  • y-intercept: \( (0, -120) \)
  • vertex: \( (1, -121) \)