characteristics of a quadratic Look at this equation $$ f(x)=-\frac{1}{3}(x+2)^{2}+3 $$ What is the vertex $$ \square $$ StretchV compress/ none $$ \square $$ is it refected $$ \square $$ Vertical Shff $$ \square $$ Horizontal shiff $$ \square $$ xintercept $$ \square $$ $$ y $$-intercept $$ \square $$ Doman $$ \square $$ Range $$ \square $$ Max/Mn $$ \square $$ $$ \mathrm{f}(1)= $$ $$ \square $$ As $$ x $$ goes to einfinty $$ \square $$ As $$ x $$ goes to infinity $$ \square $$

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Let's analyze the quadratic function $$ f(x) = -\frac{1}{3}(x + 2)^2 + 3 $$ by filling in the characteristics:

1. **Vertex**: 
   $$
   \boxed{(-2,\, 3)}
   $$
   - The vertex form of a quadratic is $$ f(x) = a(x - h)^2 + k $$, where $$(h, k)$$ is the vertex.

2. **Vertical Stretch/Compress/None**:
   $$
   \boxed{\text{Compressed vertically by } \frac{1}{3}}
   $$
   - The coefficient $$ a = -\frac{1}{3} $$ indicates a vertical compression by a factor of $$\frac{1}{3}$$.

3. **Is it Reflected?**:
   $$
   \boxed{\text{Yes, it is reflected over the } x\text{-axis}}
   $$
   - Since $$ a $$ is negative, the parabola is reflected over the $$ x $$-axis.

4. **Vertical Shift**:
   $$
   \boxed{+3}
   $$
   - The $$ +3 $$ shifts the parabola upward by 3 units.

5. **Horizontal Shift**:
   $$
   \boxed{-2}
   $$
   - The expression $$ (x + 2) $$ indicates a horizontal shift of 2 units to the **left**.

6. **$$ x $$-Intercepts**:
   $$
   \boxed{x = 1 \text{ and } x = -5}
   $$
   - To find the $$ x $$-intercepts, set $$ f(x) = 0 $$:
     $$
     -\frac{1}{3}(x + 2)^2 + 3 = 0 \implies (x + 2)^2 = 9 \implies x = 1 \text{ or } x = -5
     $$

7. **$$ y $$-Intercept**:
   $$
   \boxed{\frac{5}{3}}
   $$
   - Evaluate $$ f(0) $$:
     $$
     f(0) = -\frac{1}{3}(0 + 2)^2 + 3 = -\frac{4}{3} + 3 = \frac{5}{3}
     $$

8. **Domain**:
   $$
   \boxed{(-\infty,\, \infty)}
   $$
   - Quadratic functions have a domain of all real numbers.

9. **Range**:
   $$
   \boxed{(-\infty,\, 3]}
   $$
   - Since the parabola opens downward, the maximum $$ y $$-value is 3.

10. **Maximum/Minimum**:
    $$
    \boxed{\text{Maximum value of } 3}
    $$
    - The vertex $$ ( -2, 3 ) $$ represents the maximum point.

11. **$$ f(1) $$**:
    $$
    \boxed{0}
    $$
    - Evaluate $$ f(1) $$:
      $$
      f(1) = -\frac{1}{3}(1 + 2)^2 + 3 = -\frac{9}{3} + 3 = 0
      $$

12. **As $$ x $$ Approaches $$-\infty$$**:
    $$
    \boxed{f(x) \to -\infty}
    $$
    - The parabola decreases without bound as $$ x $$ approaches $$-\infty$$.

13. **As $$ x $$ Approaches $$ +\infty $$**:
    $$
    \boxed{f(x) \to -\infty}
    $$
    - Similarly, the parabola decreases without bound as $$ x $$ approaches $$ +\infty $$.

### Summary:

- **Vertex**: $$(-2,\, 3)$$
- **Vertical Stretch/Compress**: Compressed vertically by $$\frac{1}{3}$$
- **Reflected**: Yes, over the $$ x $$-axis
- **Vertical Shift**: +3
- **Horizontal Shift**: -2
- **$$ x $$-Intercepts**: $$ x = 1 $$ and $$ x = -5 $$
- **$$ y $$-Intercept**: $$ \frac{5}{3} $$
- **Domain**: $$ (-\infty,\, \infty) $$
- **Range**: $$ (-\infty,\, 3] $$
- **Maximum Value**: 3
- **$$ f(1) $$**: 0
- **As $$ x \to -\infty $$**: $$ f(x) \to -\infty $$
- **As $$ x \to +\infty $$**: $$ f(x) \to -\infty $$
**Characteristics of the Quadratic Function $$ f(x) = -\frac{1}{3}(x + 2)^2 + 3 $$:** - **Vertex**: $$(-2, 3)$$ - **Vertical Stretch/Compress**: Compressed vertically by $$\frac{1}{3}$$ - **Reflected**: Yes, over the $$ x $$-axis - **Vertical Shift**: +3 - **Horizontal Shift**: -2 - **$$ x $$-Intercepts**: $$ x = 1 $$ and $$ x = -5 $$ - **$$ y $$-Intercept**: $$ \frac{5}{3} $$ - **Domain**: All real numbers ($$ (-\infty, \infty) $$) - **Range**: $$ (-\infty, 3] $$ - **Maximum Value**: 3 - **$$ f(1) $$**: 0 - **As $$ x \to -\infty $$**: $$ f(x) \to -\infty $$ - **As $$ x \to +\infty $$**: $$ f(x) \to -\infty $$

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