Question

Question
What are the key points on the graph of y=x2−2x−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-intercept: (0,−120)

y-intercept: (−10,0)
(12,0)

vertex: (1,−121)

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

To analyze the quadratic function \( y = x^2 - 2x - 120 \), we need to find the vertex, x-intercepts, and y-intercept.

**1. Vertex:**
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 1 \) and \( b = -2 \):
\[
x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1
\]
Now, substitute \( x = 1 \) 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, we set \( y = 0 \):
\[
0 = x^2 - 2x - 120
\]
Factoring the quadratic:
\[
0 = (x - 12)(x + 10)
\]
Setting each factor to zero gives:
\[
x - 12 = 0 \quad \Rightarrow \quad x = 12
\]
\[
x + 10 = 0 \quad \Rightarrow \quad x = -10
\]
Thus, 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 \( (0, -120) \).

**Final answer:**
- Vertex: \( (1, -121) \)
- X-intercepts: \( (-10, 0) \) and \( (12, 0) \)
- Y-intercept: \( (0, -120) \)

Based on the provided responses, the correct choice with accurate descriptions is:
- x-intercepts: \( (-10, 0), (12, 0) \)
- y-intercept: \( (0, -120) \)
- vertex: \( (1, -121) \)