Question

consider the quadratic function f(x)=x^2-7x-8
its vertex is ( , )
its largest x-intercept is =
its y intercept is y=

Answer 1

To analyze the quadratic function \( f(x) = x^2 - 7x - 8 \), we can find the vertex, the x-intercepts, and the y-intercept.

1. **Vertex**: The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).

Here, \( a = 1 \) and \( b = -7 \):
\[
x = -\frac{-7}{2 \cdot 1} = \frac{7}{2} = 3.5
\]

To find the y-coordinate of the vertex, we substitute \( x = 3.5 \) back into the function:
\[
f(3.5) = (3.5)^2 - 7(3.5) - 8
\]
\[
= 12.25 - 24.5 - 8
\]
\[
= 12.25 - 32.5 = -20.25
\]
So, the vertex is \( \left( 3.5, -20.25 \right) \).

2. **X-intercepts**: To find the x-intercepts, we set \( f(x) = 0 \):
\[
x^2 - 7x - 8 = 0
\]
We can factor this quadratic:
\[
(x - 8)(x + 1) = 0
\]
This gives us the x-intercepts:
\[
x - 8 = 0 \quad \Rightarrow \quad x = 8
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
The largest x-intercept is \( x = 8 \).

3. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
\[
f(0) = 0^2 - 7(0) - 8 = -8
\]
Thus, the y-intercept is \( y = -8 \).

### Summary
- The vertex is \( (3.5, -20.25) \)
- The largest x-intercept is \( 8 \)
- The y-intercept is \( y = -8 \)