Question

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

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**: \( [-3, \infty) \)