Question

What are the vertex and x-intercepts of the graph of the function given below?
y=x2-2x-24
• Vertex: (1, -25); x-intercepts: (6, 0) and (-4, 0)
• Vertex: (1, -21); x-intercepts: (6, 0) and (4, 0)
Vertex: (0, 0); x-intercepts: (-4, 0) and (6, 0)
Vertex: (7,5); x-intercepts: (6, 0) and (8, 0)

Answer 1

To find the vertex and x-intercepts of the quadratic function \( y = x^2 - 2x - 24 \), we can use the following formulas and methods.

1. **Finding the Vertex**: The x-coordinate of the vertex \( x_v \) can be found using the formula:
\[
x_v = -\frac{b}{2a}
\]
Here \( a = 1 \) and \( b = -2 \).
\[
x_v = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1
\]
Now, we can find the y-coordinate of the vertex by substituting \( x_v \) back into the equation:
\[
y_v = (1)^2 - 2(1) - 24 = 1 - 2 - 24 = -25
\]
So, the vertex is \( (1, -25) \).

2. **Finding the x-intercepts**: The x-intercepts occur when \( y = 0 \):
\[
0 = x^2 - 2x - 24
\]
We can factor this quadratic:
\[
0 = (x - 6)(x + 4)
\]
Setting each factor equal to zero gives us the x-intercepts:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]
Thus, the x-intercepts are \( (6, 0) \) and \( (-4, 0) \).

In summary, the vertex is \( (1, -25) \) and the x-intercepts are \( (6, 0) \) and \( (-4, 0) \).

Thus, the correct answer is:
**Vertex: (1, -25); x-intercepts: (6, 0) and (-4, 0)**.