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analyse this quadratic function (domain, range,etc)

The equation provided is:

\[
y = 4 - x^2
\]

This is a quadratic function. Let's analyze its key features:

### 1. **Standard Form**:
The equation is already in the standard form of a quadratic function:

\[
y = -x^2 + 4
\]

This represents a parabola that opens **downwards** because the coefficient of \(x^2\) is negative.

### 2. **Vertex**:
The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola.

In this case, the equation can be rewritten as:

\[
y = -(x - 0)^2 + 4
\]

So, the vertex is at \((0, 4)\).

### 3. **Axis of Symmetry**:
The axis of symmetry is a vertical line that passes through the vertex. Since the vertex is at \((0, 4)\), the axis of symmetry is:

\[
x = 0
\]

### 4. **Domain**:
For any quadratic function, the domain is all real numbers, because \(x\) can take any value.

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

### 5. **Range**:
Since the parabola opens downwards and the vertex is the maximum point of the function (at \(y = 4\)), the function will take values less than or equal to 4. Therefore, the range is:

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

### 6. **Intercepts**:
- **Y-intercept**: Set \(x = 0\):
  \[
  y = 4 - 0^2 = 4
  \]
  So, the y-intercept is \((0, 4)\).

- **X-intercepts**: Set \(y = 0\):
  \[
  0 = 4 - x^2
  \]
  Solving for \(x\):
  \[
  x^2 = 4 \quad \Rightarrow \quad x = \pm 2
  \]
  So, the x-intercepts are \((2, 0)\) and \((-2, 0)\).

### 7. **Concavity**:
Since the coefficient of \(x^2\) is negative, the parabola is concave down.

---

Would you like more details on any part of this analysis or another related topic? Here are some related questions you could explore:

1. How do you convert a quadratic function to vertex form?
2. What are the effects of the coefficient of \(x^2\) on the shape of the parabola?
3. Can you explain the process of finding the axis of symmetry in different quadratics?
4. How do transformations affect the graph of a quadratic function?
5. What are the applications of quadratic functions in real life?

**Tip**: Always use the discriminant \(b^2 - 4ac\) to determine the number of real solutions when solving quadratic equations.

Analyze this quadratic function (domain, range, etc.)

Explore the analysis of the quadratic equation y = 4 - x^2, including its domain, range, vertex, intercepts, and symmetry. Learn key concepts like vertex form and the concavity of the parabola. Suitable for students in Grades 8-10.

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