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 given equation: $x=-4\left(y-1\right)2^{}+3$. This is a parabola expressed in terms of $y$.
2. Recognize the vertex form of the parabola. Here, the vertex is at $(3,\; 1)$ because the equation is in the form $x\; =\; a(y\; -\; k)^2\; +\; h$, where $(h,\; k)$ is the vertex.
3. Determine the direction the parabola opens by looking at the coefficient of the squared term, $a\; =\; -4$. Since $a$ is negative, the parabola opens to the left (towards decreasing $x$ values).
4. Find the domain by considering the range of $x$ values. Since the parabola opens left from the vertex at $x\; =\; 3$, the domain is all $x$ values less than or equal to 3, or $x\; \backslash leq\; 3$.
5. Find the range by considering all possible $y$ values. Because the parabola is symmetric about $y\; =\; 1$ and opens horizontally, $y$ can take any real value, so the range is $(-\backslash infty,\; \backslash infty)$. Finally, determine if the relation is a function: since for some $x$ values there are multiple $y$ values, it is not a function.

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola expresses the equation in a way that reveals its vertex, the highest or lowest point. For example, x = a(y - k)^2 + h shows a parabola with vertex at (h, k). Understanding the vertex helps identify key features like the parabola's position and symmetry.

Direction of Opening of a Parabola

The sign and variable squared in the equation determine the parabola's direction. If the squared term is y, the parabola opens horizontally; if x, it opens vertically. The coefficient's sign indicates whether it opens left/right or up/down, which affects the domain and range.

Domain and Range of Relations and Functions

The domain is the set of all possible input values (x or y), and the range is the set of all possible output values. For parabolas, these depend on the vertex and opening direction. Determining if the relation is a function involves checking if each input corresponds to exactly one output.