Question

The quadratic function f of x equals negative 3 x squared plus 12 x minus 7 can be rewritten in vertex form to reveal the vertex of the function.



Move the numbers to the blanks to show the vertex form and identify the vertex of f of x.

Response area with 4 blank spaces
f of x equals negative 3 left-parenthesis x minus
Blank space 1 empty
right-parenthesis squared plus
Blank space 2 empty


The vertex of the function is at the point left-parenthesis
Blank space 3 empty
comma
Blank space 5 empty
right-parenthesis.

Answer 1

To rewrite the quadratic function \( f(x) = -3x^2 + 12x - 7 \) in vertex form, we can follow these steps:

1. **Factor out the coefficient of \( x^2 \)** from the first two terms.
2. **Complete the square** inside the parentheses.
3. **Write the function in vertex form**.

Starting with the function:

\[
f(x) = -3x^2 + 12x - 7
\]

1. **Factor out -3 from the first two terms**:

\[
f(x) = -3(x^2 - 4x) - 7
\]

2. **Complete the square**:

- Take half of the coefficient of \( x \) (which is -4), square it: \( (-4/2)^2 = 4 \).
- Add and subtract this value inside the parentheses.

\[
f(x) = -3(x^2 - 4x + 4 - 4) - 7
\]

\[
f(x) = -3((x - 2)^2 - 4) - 7
\]

3. **Distributing -3**:

\[
f(x) = -3(x - 2)^2 + 12 - 7
\]

\[
f(x) = -3(x - 2)^2 + 5
\]

Now we have the function in vertex form:

\[
f(x) = -3(x - 2)^2 + 5
\]

### Filling the Blanks:

1. **Blank space 1**: 2 (from \( x - 2 \))
2. **Blank space 2**: 5 (the constant term after completing the square)
3. **Blank space 3**: 2 (x-coordinate of the vertex)
4. **Blank space 4**: 5 (y-coordinate of the vertex)

Thus, the completed response is:

\[
f(x) = -3(x - 2)^2 + 5
\]

The vertex of the function is at the point \((2, 5)\).

Here are the blanks filled in:

- f of x equals negative 3 left-parenthesis x minus **2** right-parenthesis squared plus **5**
- The vertex of the function is at the point left-parenthesis **2** comma **5** right-parenthesis.