Problem 53 - College Algebra

Question

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-4(y-1)^{2}+3$$

Step-by-Step Solution

Answer

The parabola opens to the left. The vertex is (3,1). The domain of the given relation is \(-\infty, 3\] and the range is (-\infty, \infty). This relation is not a function.

Step 1: Identify the Direction of the Parabolic Opening

The parabola equation is given in the form \(x = a(y-k)^2 + h\), which indicates that the graph opens either to the left if \(a < 0\) or to the right if \(a > 0\). In the given equation, \(x = -4(y-1)^2 + 3\), 'a' is -4 which is less than 0, hence, the parabola opens to the left.

Step 2: Calculate the Domain and Range

Since the parabola opens to the left and not up or down, the domain and range are different than usual. The domain (x-values) includes all real numbers less than or equal to the x-coordinate of the vertex because it opens to the left. Hence, the domain is \(-\infty, 3\]. The range (y-values), on the other hand, is all real numbers (-\infty, \infty) because the vertex's y-coordinate doesn't limit the graph.

Step 3: Determine Whether the Relation is a Function

The relation is not a function. For every x-value in the domain, there are two corresponding y-values (above and below the vertex), invoking the 'vertical line test'. This violates the condition for a relation to be a function as each input (x-value) should have a unique output (y-value).