5. $$ y=-3 x^{2}+6 x+4 $$ domain: range:

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To determine the **domain** and **range** of the quadratic function:

$$ y = -3x^{2} + 6x + 4 $$

let's analyze each separately.

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### **1. Domain**

The **domain** of a function consists of all possible input values ($$ x $$) for which the function is defined.

- **Quadratic functions** are defined for all real numbers since you can substitute any real number for $$ x $$ and compute a corresponding $$ y $$.

**Therefore, the domain is:**

$$ \boxed{(-\infty,\ \infty)} $$

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### **2. Range**

The **range** of a function consists of all possible output values ($$ y $$) that the function can produce.

For the given quadratic function:

$$ y = -3x^{2} + 6x + 4 $$

- The coefficient of $$ x^{2} $$ is **-3**, which is **negative**. This means the parabola **opens downward**, indicating that there is a **maximum** point (vertex) on the graph.

**To find the range, follow these steps:**

1. **Find the vertex of the parabola.**

The vertex form of a quadratic function $$ y = ax^{2} + bx + c $$ has its vertex at:

$$ x = -\frac{b}{2a} $$

Plugging in the coefficients:

$$ a = -3, \quad b = 6 $$
   
   $$ x = -\frac{6}{2(-3)} = -\frac{6}{-6} = 1 $$

2. **Determine the maximum $$ y $$-value by substituting $$ x = 1 $$ back into the original equation.**

$$ y = -3(1)^2 + 6(1) + 4 = -3 + 6 + 4 = 7 $$

3. **Interpret the range based on the vertex.**

Since the parabola opens downward and the maximum $$ y $$-value is 7, the function will produce all real numbers **less than or equal to 7**.

**Therefore, the range is:**

$$ \boxed{(-\infty,\ 7]} $$

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### **Summary**

- **Domain:** $$ (-\infty,\ \infty) $$
- **Range:** $$ (-\infty,\ 7] $$
The domain is all real numbers, and the range is all real numbers less than or equal to 7.

5. \( y=-3 x^{2}+6 x+4 \) domain: range:

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