Question

What are the vertex and x-intercepts of the graph of the function given below?
y=x2-2x-35
Vertex: (7, 5); x-intercepts: (7, 0) and (8, 0)
Vertex: (0, 0); x-intercepts: (-4, 0) and (-5, 0)
Vertex: (1, -36); x-intercepts: (7, 0) and (-5, 0)
Vertex: (1, -25); x-intercepts: (6, 0) and (-4, 0)

Answer 1

To find the vertex and x-intercepts of the function \( y = x^2 - 2x - 35 \), we can follow these steps:

1. **Finding the Vertex**:
The x-coordinate of the vertex of a quadratic function in the standard form \( y = ax^2 + bx + c \) can be found using the formula:

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

In our case, \( a = 1 \) and \( b = -2 \).

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

Now, we substitute \( x_v = 1 \) back into the function to find the y-coordinate of the vertex:

\[
y_v = (1)^2 - 2(1) - 35 = 1 - 2 - 35 = -36
\]

So, the vertex is \( (1, -36) \).

2. **Finding the x-intercepts**:
The x-intercepts occur where \( y = 0 \):

\[
0 = x^2 - 2x - 35
\]

We can factor this quadratic:

\[
0 = (x - 7)(x + 5)
\]

Setting each factor to zero gives:

\[
x - 7 = 0 \quad \Rightarrow \quad x = 7
\]
\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\]

Therefore, the x-intercepts are \( (7, 0) \) and \( (-5, 0) \).

### Summary

- Vertex: \( (1, -36) \)
- x-intercepts: \( (7, 0) \) and \( (-5, 0) \)

**Based on your options**, the correct choice is:

**Vertex: (1, -36); x-intercepts: (7, 0) and (-5, 0)**.