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

Answer 1

To find the vertex of the quadratic function \( f(x) = 6x^2 + 12x - 7 \), we can use the formula for the vertex given by \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the standard form \( ax^2 + bx + c \).

### Step 1: Identify coefficients
In the given function,
- \( a = 6 \)
- \( b = 12 \)
- \( c = -7 \)

### Step 2: Calculate the x-coordinate of the vertex
Using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{12}{2 \cdot 6} = -\frac{12}{12} = -1
\]

### Step 3: Calculate the y-coordinate of the vertex
To find the y-coordinate of the vertex, substitute \( x = -1 \) back into the function \( f(x) \):
\[
f(-1) = 6(-1)^2 + 12(-1) - 7
\]
\[
= 6(1) - 12 - 7
\]
\[
= 6 - 12 - 7
\]
\[
= 6 - 19
\]
\[
= -13
\]

### Step 4: Write the vertex as an ordered pair
Therefore, the vertex is:
\[
(-1, -13)
\]

### Step 5: Determine the axis of symmetry
The axis of symmetry for a quadratic function in standard form is given by the vertical line that passes through the vertex. From our previous calculation, the x-coordinate of the vertex is \( -1 \).

Thus, the axis of symmetry is:
\[
x = -1
\]

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