Question

For each of the following quadratic functions (Show your work):
Calculate the vertex by hand and write it as an ordered pair.
Determine the axis of symmetry and write it as a linear equation (x = # or t = #).
Function g(x)x^2/2-2x-9
-b/2a
f(-b/2a)
vertex
axis of symmetry

Answer 1

Let's calculate the vertex and the axis of symmetry for the quadratic function \( g(x) = \frac{x^2}{2} - 2x - 9 \).

### Step 1: Identify coefficients

The function is of the form \( g(x) = ax^2 + bx + c \), where:
- \( a = \frac{1}{2} \)
- \( b = -2 \)
- \( c = -9 \)

### Step 2: Calculate the x-coordinate of the vertex using the formula \( x = -\frac{b}{2a} \)

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{b}{2a} = -\frac{-2}{2 \cdot \frac{1}{2}} = \frac{2}{1} = 2
\]

### Step 3: Calculate the y-coordinate of the vertex by substituting \( x = 2 \) back into the function

Now we will find \( g(2) \):

\[
g(2) = \frac{(2)^2}{2} - 2(2) - 9
\]
\[
= \frac{4}{2} - 4 - 9
\]
\[
= 2 - 4 - 9 = -11
\]

### Step 4: Write the vertex as an ordered pair

The vertex of the function \( g(x) \) is:

\[
\text{Vertex} = (2, -11)
\]

### Step 5: Write the axis of symmetry

The axis of symmetry can be expressed with the equation:

\[
x = 2
\]

### Summary of Results

- **Vertex:** \( (2, -11) \)
- **Axis of Symmetry:** \( x = 2 \)