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

Question

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: \( (-\infty, 0] \)