In Exercises 57–62, 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

Verified step by step guidance

1. Identify the vertex of the parabola from the equation x = -4(y - 1)^2 + 3. The vertex form of a parabola in terms of y is x = a(y - k)^2 + h, where (h, k) is the vertex. Comparing, we find the vertex is (3, 1).
2. Determine the direction in which the parabola opens. Since the coefficient of the squared term, -4, is negative, the parabola opens to the left.
3. Establish the domain of the parabola. Since the parabola opens leftward, all x-values are less than or equal to the x-coordinate of the vertex. Therefore, the domain is all x such that x ≤ 3.
4. Establish the range of the parabola. Since the parabola is a sideways parabola (opening left or right) and not restricted vertically, the range is all real numbers, or (-∞, ∞).
5. Determine if the relation is a function. A relation is a function if each input (x-value) corresponds to exactly one output (y-value). Since this parabola is sideways, for some x-values, there are multiple corresponding y-values. Therefore, this relation is not a function.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. For the equation given, the vertex can be identified from the standard form of a parabola, which is expressed as x = a(y - k)^2 + h, where (h, k) is the vertex. In this case, the vertex is at the point (3, 1).

Direction of Opening

The direction in which a parabola opens is determined by the coefficient of the squared term. If the coefficient is positive, the parabola opens upwards; if negative, it opens downwards. In the provided equation, the coefficient is -4, indicating that the parabola opens to the left.

Domain and Range

The domain of a function refers to all possible input values (x-values), while the range refers to all possible output values (y-values). For the given parabola, since it opens to the left, the domain is all real numbers, while the range is limited to y-values greater than or equal to 1, as the vertex represents the maximum point.