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: (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

x-intercepts: (−10,0) (12,0)

y-intercept: (1,−121)

vertex: (0,−120)

Answer 1

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

1. **Vertex:**
The vertex form of a quadratic equation \( y = ax^2 + bx + c \) has its x-coordinate given by \( x = -\frac{b}{2a} \).
For the equation \( y = 1x^2 - 2x - 120 \):
- \( a = 1 \)
- \( b = -2 \)
- \( x = -\frac{-2}{2(1)} = 1 \)

To find the y-coordinate of the vertex, substitute \( x = 1 \) back into the equation:
\[
y = (1)^2 - 2(1) - 120 = 1 - 2 - 120 = -121
\]
So, the vertex is \( (1, -121) \).

2. **X-Intercepts:**
To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 2x - 120
\]
Factoring or using the quadratic formula, we find:
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-120)}}{2(1)} = \frac{2 \pm \sqrt{4 + 480}}{2} = \frac{2 \pm \sqrt{484}}{2}
\]
\[
\sqrt{484} = 22 \Rightarrow x = \frac{2 \pm 22}{2}
\]
This gives:
\[
x = \frac{24}{2} = 12 \quad \text{and} \quad x = \frac{-20}{2} = -10
\]
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 \( (0, -120) \).

In summary, the key points of the graph are:
- **Vertex:** \( (1, -121) \)
- **X-intercepts:** \( (-10, 0) \) and \( (12, 0) \)
- **Y-intercept:** \( (0, -120) \)

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