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

Question

f, of, x, equals, minus, start fraction, 2, divided by, 5, end fraction, left parenthesis, x, minus, 3, right parenthesis, squared, plus, 4f(x)=−
5
2
​
(x−3)
2
+4

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 analyze the given parabola described by the equation:

\[ f(x) = -\frac{2}{5}(x - 3)^2 + 4 \]

1. Vertex:

The vertex form of a parabola is given by:

\[ f(x) = a(x - h)^2 + k \]

where \((h, k)\) is the vertex. In your equation:

• \(h = 3\)
• \(k = 4\)

So, the vertex is:

\[ \text{Vertex: } (3, 4) \]

2. Maximum or Minimum Point:

Since the coefficient of the squared term (\(a = -\frac{2}{5}\)) is negative, the parabola opens downwards. This means that the vertex represents a maximum point.

So, the answer is:

\[ \text{Maximum point: } (3, 4) \]

3. Axis of Symmetry:

The axis of symmetry of a parabola in vertex form is the vertical line that passes through the vertex. It can be represented as:

\[ x = h \]

From the vertex we found \(h = 3\), therefore:

\[ \text{Axis of Symmetry: } x = 3 \]

4. Domain:

The domain of any quadratic function is all real numbers, so:

\[ \text{Domain: } (-\infty, \infty) \]

5. Range:

Since the parabola opens downward and the maximum value occurs at the vertex \(y = 4\), the range starts from negative infinity and goes up to the maximum \(y\)-value:

\[ \text{Range: } (-\infty, 4] \]

Summary:

• Vertex: (3, 4)
• Maximum or Minimum Point: Maximum point (3, 4)
• Axis of Symmetry: x = 3
• Domain: (-∞, ∞)
• Range: (-∞, 4]