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Chapter 3: Problem 46

Give the domain and the range of each quadratic function whose graph is described. The vertex is \((-3,-4)\) and the parabola opens down.

Short Answer

Expert verified
The domain of the function is \((-鈭�, 鈭�)\) and the range is \((-鈭�, -4]\)

Step by step solution

01

Identifying the vertex

The vertex mentioned in the problem is \((-3, -4)\), providing the maximum value for the function's range (since the parabola opens downwards). This information is crucial to determine the range.
02

Determining the direction of the parabola

The problem says the parabola opens downwards. If a parabola opens downwards, the highest point it reaches is the y-value of the vertex, and it continues indefinitely below this. This implies that the y-values extend to negative infinity.
03

Defining the domain of the function

For any quadratic function, the domain is all real numbers, because you can plug any real number into the function and get a valid output. So, the domain is \((-鈭�, 鈭�)\).
04

Defining the range of the function

Given the vertex and the direction of the parabola, the range of the function will be from the y-coordinate of the vertex to negative infinity, because the parabola opens downwards. Hence, the range is \((-鈭�, -4]\).

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