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 f(x)=6x^2+12x-7
-b/2a
f(-b/2a)
vertex
axis of symmetry

Answer 1

To find the vertex and axis of symmetry of the quadratic function \( f(x) = 6x^2 + 12x - 7 \), we can follow these steps:

1. **Identify the coefficients** from the standard form of a quadratic function \( ax^2 + bx + c \):
- Here, \( a = 6 \), \( b = 12 \), and \( c = -7 \).

2. **Calculate the x-coordinate of the vertex** using the formula:
\[
x = -\frac{b}{2a}
\]

Substituting in the values of \( b \) and \( a \):
\[
x = -\frac{12}{2 \cdot 6} = -\frac{12}{12} = -1
\]

3. **Calculate the y-coordinate of the vertex** by substituting \( x = -1 \) back into the original function \( f(x) \):
\[
f(-1) = 6(-1)^2 + 12(-1) - 7
\]
\[
= 6(1) - 12 - 7
\]
\[
= 6 - 12 - 7
\]
\[
= -13
\]

4. **Write the vertex** as an ordered pair:
\[
\text{Vertex} = (-1, -13)
\]

5. **Determine the axis of symmetry**: The axis of symmetry is a vertical line that passes through the vertex. It can be written as:
\[
x = -1
\]

### Summary of Results:
- **Vertex**: \( (-1, -13) \)
- **Axis of Symmetry**: \( x = -1 \)