Question

Identify the characteristics of the parabola with the given equation below:

f, of, x, equals, minus, left parenthesis, x, minus, 1, right parenthesis, squaredf(x)=−(x−1)
2


Answer
Attempt 1 out of 3

What is the vertex?


Is the vertex a maximum point or a minimum point?


What is the axis of symmetry?


Domain:


Range:

Answer 1

Let's analyze the given function:

\[ f(x) = -(x - 1)^2 \]

### Characteristics of the Parabola:

1. **Vertex**:
The vertex of the parabola can be found from the equation in vertex form, which is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. For this equation, we can see that:
- \( h = 1 \)
- \( k = 0 \)
Thus, the vertex is:
\[
\text{Vertex} = (1, 0)
\]

2. **Maximum or Minimum Point**:
The coefficient \( a \) (in front of the squared term) is negative (-1). This indicates that the parabola opens downward. Therefore, the vertex is a maximum point. Thus:
\[
\text{Maximum Point}
\]

3. **Axis of Symmetry**:
The axis of symmetry for a parabola in vertex form is given by the line \( x = h \). In this case:
\[
\text{Axis of Symmetry} = x = 1
\]

4. **Domain**:
The domain of a parabola that opens downwards is all real numbers. Hence:
\[
\text{Domain} = (-\infty, \infty)
\]

5. **Range**:
Since the vertex is the maximum point at \( y = 0 \) and the parabola opens downward, the range includes all values less than or equal to 0. Thus:
\[
\text{Range} = (-\infty, 0]
\]

### Summary:
- **Vertex**: \( (1, 0) \)
- **Maximum Point**: Yes, it is a maximum point.
- **Axis of Symmetry**: \( x = 1 \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 0] \)