Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The domain of the function is \((-∞, ∞)\) and the range is \((-∞, -4]\)

Step 1: 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.

Step 2: 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.

Step 3: 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 \((-∞, ∞)\).

Step 4: 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 \((-∞, -4]\).