Determine the key features of the Quadratic Function y=-(x-2)^2+4

To determine the key features of the quadratic function \( y = -(x - 2)^2 + 4 \), we need to analyze its components and behavior. This equation is in the vertex form of a quadratic function, which is generally written as:

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

where:
- \( (h, k) \) is the **vertex** of the parabola,
- \( a \) determines the **direction** and **width** of the parabola.

### 1. **Vertex**:
The vertex of the function \( y = -(x - 2)^2 + 4 \) can be directly identified from the equation. The vertex form tells us that the vertex is at the point \( (h, k) \), where \( h = 2 \) and \( k = 4 \). Hence, the vertex is:

\[
\text{Vertex} = (2, 4)
\]

### 2. **Axis of Symmetry**:
The axis of symmetry is the vertical line that passes through the vertex. For a quadratic function in vertex form, the axis of symmetry is \( x = h \). Therefore, the axis of symmetry for this function is:

\[
\text{Axis of Symmetry} = x = 2
\]

### 3. **Direction of Opening**:
The coefficient of \( (x - 2)^2 \) is negative (since it's \( -(x - 2)^2 \)). This tells us that the parabola opens **downward**.

### 4. **Maximum or Minimum**:
Since the parabola opens downward, the vertex represents the **maximum** value of the function. The maximum value is the \( y \)-coordinate of the vertex, which is 4. Thus, the maximum value of the function is:

\[
\text{Maximum} = 4
\]

### 5. **Y-Intercept**:
To find the y-intercept, we substitute \( x = 0 \) into the equation and solve for \( y \):

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

Thus, the y-intercept is:

\[
\text{Y-Intercept} = (0, 0)
\]

### 6. **X-Intercepts**:
To find the x-intercepts, we set \( y = 0 \) and solve for \( x \):

\[
0 = -(x - 2)^2 + 4
\]
\[
(x - 2)^2 = 4
\]
\[
x - 2 = \pm 2
\]
\[
x = 2 \pm 2
\]
\[
x = 4 \quad \text{or} \quad x = 0
\]

Thus, the x-intercepts are:

\[
\text{X-Intercepts} = (0, 0) \text{ and } (4, 0)
\]

### Summary of Key Features:
1. **Vertex**: \( (2, 4) \)
2. **Axis of Symmetry**: \( x = 2 \)
3. **Direction of Opening**: Downward
4. **Maximum Value**: 4
5. **Intercepts**: X-Intercepts \( (0, 0) \) and \( (4, 0) \), Y-Intercept \( (0, 0) \)

Would you like further details or have any questions?

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Here are 5 related questions:
1. How would the function change if the coefficient of \( (x-2)^2 \) were positive?
2. What effect would shifting the function up or down have on its maximum value?
3. How can you find the domain and range of this quadratic function?
4. What is the impact of horizontal shifts on the vertex of a quadratic function?
5. How do you solve for the vertex form of a quadratic function if it's given in standard form?

**Tip**: The sign of the leading coefficient determines whether the parabola opens upwards (positive) or downwards (negative).

Explore the key features of the quadratic function y=-(x-2)^2+4, including its vertex at (2, 4), axis of symmetry at x=2, and intercepts. This detailed explanation covers essential concepts of quadratic functions, suitable for Grades 8-10.

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