What is the range of the quadratic function whose equation I s `q(x)=-(x+4)^(2)+3`?

Options

A. `yge-4`
B. `yle3`
C. `yge3`
D. `yle4`

Explanation

To find the range of the quadratic function given by the equation \( q(x) = -(x + 4)^2 + 3 \), we can follow these steps:

### Step 1: Identify the form of the quadratic function
The function is in the vertex form of a quadratic equation, which is given by:
\[ q(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In our case, we can rewrite the function as:
\[ q(x) = -1(x - (-4))^2 + 3 \]
Here, \( a = -1 \), \( h = -4 \), and \( k = 3 \).

### Step 2: Determine the vertex
From the vertex form, we can see that the vertex of the parabola is at the point \((-4, 3)\). This is the highest point of the parabola since \( a < 0 \) (the parabola opens downwards).

### Step 3: Analyze the direction of the parabola
Since \( a = -1 \) is negative, the parabola opens downwards. This means that the maximum value of \( q(x) \) occurs at the vertex, which is \( k = 3 \).

### Step 4: Determine the range
The range of the function is the set of all possible output values \( q(x) \) can take. Since the parabola opens downwards and the maximum value is \( 3 \), the function can take any value less than or equal to \( 3 \). Therefore, the range of \( q(x) \) is:
\[ q(x) \leq 3 \]
In interval notation, this can be expressed as:
\[ (-\infty, 3] \]

### Final Answer
The range of the quadratic function \( q(x) = -(x + 4)^2 + 3 \) is:
\[ (-\infty, 3] \]

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