Question

1. $y = -\frac{1}{2}(x - 1)(x - 9)$

p = 1 q = 9
vértice: x = \frac{p + q}{2} = \frac{1 + 9}{2}$
$y = -\frac{1}{2}(x - 1)(x - 9)$
$= -\frac{1}{2}(5 - 1)(5 - 9)$
$= -112.5$ $(5, -112.5)$

Explanation:

Step 1: Find the x - coordinate of the vertex

For a quadratic function in factored form \( y=a(x - p)(x - q) \), the x - coordinate of the vertex (the axis of symmetry) is given by the formula \( x=\frac{p + q}{2} \).
Here, \( p = 1 \) and \( q = 9 \), so \( x=\frac{1+9}{2}=\frac{10}{2}=5 \).

Step 2: Find the y - coordinate of the vertex

Substitute \( x = 5 \) into the function \( y=-\frac{1}{2}(x - 1)(x - 9) \).
First, calculate \( (x - 1) \) and \( (x - 9) \) when \( x = 5 \):
\( x-1=5 - 1 = 4 \) and \( x - 9=5 - 9=-4 \).
Then, \( y=-\frac{1}{2}(4)\times(-4) \).
\( y=-\frac{1}{2}\times(-16) \).
\( y = 8 \).

So the vertex of the parabola \( y = -\frac{1}{2}(x - 1)(x - 9) \) is \( (5, 8) \).

Answer:

The vertex of the parabola \( y = -\frac{1}{2}(x - 1)(x - 9) \) is \( (5, 8) \) (Note: There was a miscalculation in the original handwritten work. Let's correct it step - by - step).