Question

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

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]