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

Question

f, of, x, equals, one half, left parenthesis, x, minus, 7, right parenthesis, squared, minus, 3f(x)=
2
1
​
(x−7)
2
−3

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

To identify the characteristics of the given parabola \( f(x) = \frac{1}{2} (x - 7)^2 - 3 \), we can analyze it using the standard form of a quadratic function.

1. Vertex: The function is in the vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the equation, we see:

   • \( h = 7 \)
   • \( k = -3 \) So the vertex is \( (7, -3) \).

2. Maximum or Minimum:

   • The coefficient \( a = \frac{1}{2} \) is positive, which indicates that the parabola opens upwards. Therefore, the vertex is a minimum point.

3. Axis of Symmetry:

   • The axis of symmetry can be found from the vertex \( x = h \). Thus, the axis of symmetry is:
   • \( x = 7 \).

4. Domain:

   • The domain of a parabola is always all real numbers, which can be expressed as:
   • \( (-\infty, \infty) \).

5. Range:

   • Since the parabola opens upwards and the minimum point (vertex) is at \( y = -3 \), the range starts from this minimum value to positive infinity. Thus, the range is:
   • \( [-3, \infty) \).

Summary of the Characteristics:

• Vertex: \( (7, -3) \)
• Maximum/Minimum: Minimum
• Axis of Symmetry: \( x = 7 \)
• Domain: \( (-\infty, \infty) \)
• Range: \( [-3, \infty) \)