Question

What are the vertex and x-intercepts of the graph of y= x2 - 2x - 24?
Select one answer for the vertex and one for the x-intercepts.
O x-intercepts: (4, 0), (-6, 0)
O x-intercepts: (-4, 0), (6, 0)
• Vertex: (1, 23)
• x-intercepts: (-4, 0), (-6, 0)
O Vertex: (-1, -21)
• Vertex: (1, -25)

Answer 1

To find the vertex and x-intercepts of the quadratic function \(y = x^2 - 2x - 24\), let's start with finding the vertex.

The vertex \( (h, k) \) of a parabola given by the equation \(y = ax^2 + bx + c\) can be calculated using:

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

In our equation, \(a = 1\), \(b = -2\), and \(c = -24\).

Calculating \(h\):

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

Next, we substitute \(h\) back into the equation to find \(k\):

\[
k = (1)^2 - 2(1) - 24 = 1 - 2 - 24 = -25
\]

Thus, the vertex is \((1, -25)\).

Next, let's find the x-intercepts. X-intercepts occur when \(y = 0\):

\[
0 = x^2 - 2x - 24
\]

To solve for \(x\), we can factor the quadratic:

\[
0 = (x - 6)(x + 4)
\]

Setting each factor to zero gives:

\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
\[
x + 4 = 0 \quad \Rightarrow \quad x = -4
\]

Thus, the x-intercepts are \((6, 0)\) and \((-4, 0)\).

Now, let's summarize the results:

- Vertex: \((1, -25)\)
- X-intercepts: \((6, 0), (-4, 0)\)

Based on the choices provided in your question we have:
- The vertex is: **Vertex: (1, -25)**
- The x-intercepts are: **x-intercepts: (-4, 0), (6, 0)**

Both answers are correct based on the calculations we performed.