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-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-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-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

Answer 1

To find the key points on the graph of the function \( y = x^2 - 2x - 120 \), we can analyze the equation.

### 1. **Vertex**:
The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For this quadratic:
- \( a = 1 \)
- \( b = -2 \)

Plugging in the values:
\[
x = -\frac{-2}{2 \times 1} = 1
\]
Now, substitute \( x = 1 \) back into the equation to find the y-coordinate of the vertex:
\[
y = (1)^2 - 2(1) - 120 = 1 - 2 - 120 = -121
\]
Thus, the vertex is at the point:
\[
\text{Vertex: } (1, -121)
\]

### 2. **Y-Intercept**:
The y-intercept occurs when \( x = 0 \):
\[
y = (0)^2 - 2(0) - 120 = -120
\]
So the y-intercept is at:
\[
\text{Y-Intercept: } (0, -120)
\]

### 3. **X-Intercepts**:
To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 2x - 120
\]
This can be factored to:
\[
(x - 12)(x + 10) = 0
\]
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:
\[
\text{X-Intercepts: } (-10, 0) \text{ and } (12, 0)
\]

### Summary of Intercepts and Vertex:
- **X-Intercepts**: \((-10, 0)\) and \((12, 0)\)
- **Y-Intercept**: \((0, -120)\)
- **Vertex**: \((1, -121)\)

Therefore, the correct response is:
- x-intercepts: \((-10, 0)\) and \((12, 0)\)
- y-intercept: \((0, -120)\)
- vertex: \((1, -121)\)