Question

What are the key points on the graph of y=x2−2x−120
? Name the vertex, x-intercepts, and y-intercept. Write your answers as ordered pairs. For the x-intercepts, write the leftmost point first. (4 points)
x-intercepts:
and

y-intercept:

vertex:

Answer 1

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

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

2. **X-Intercepts:**
To find the x-intercepts, we set \(y = 0\) and solve for \(x\):
\[
0 = x^2 - 2x - 120
\]
We can factor this quadratic equation:
\[
0 = (x - 12)(x + 10)
\]
Setting each factor to zero gives us:
\[
x - 12 = 0 \quad \Rightarrow \quad x = 12
\]
\[
x + 10 = 0 \quad \Rightarrow \quad x = -10
\]
Therefore, the x-intercepts are:
\[
(-10, 0) \quad \text{and} \quad (12, 0)
\]

3. **Vertex:**
The vertex of a quadratic in the standard form \(y = ax^2 + bx + c\) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For our function, \(a = 1\) and \(b = -2\):
\[
x = -\frac{-2}{2 \cdot 1} = 1
\]
To find the corresponding \(y\)-value, we substitute \(x = 1\) back into the equation:
\[
y = 1^2 - 2(1) - 120 = 1 - 2 - 120 = -121
\]
Thus, the vertex is:
\[
(1, -121)
\]

Finally, we can summarize the key points:

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

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