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-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-intercepts: (−10,0)
(12,0)

y-intercept: (1,−121)

vertex: (0,−120)

Answer 1

To identify the key points of the quadratic function \( y = x^2 - 2x - 120 \), we can determine the vertex, x-intercepts, and y-intercept as follows:

### 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:

\[
x = -\frac{b}{2a}
\]

For the equation \( y = x^2 - 2x - 120 \):
- \( a = 1 \)
- \( b = -2 \)
- \( c = -120 \)

Calculating the x-coordinate of the vertex:

\[
x = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1
\]

Now, 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 at **(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:

1. \( x - 12 = 0 \) → \( x = 12 \)
2. \( x + 10 = 0 \) → \( x = -10 \)

Thus, the x-intercepts are at **(-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 at **(0, -120)**.

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

The correct response based on the analysis above is:

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