Problem 61 - Algebra and Trigonometry

Question

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? $$ x=-4(y-1)^{2}+3 $$

Step-by-Step Solution

Answer

The vertex is at (1, 3) and the parabola opens to the left. The domain of the relation is \[-\infty, 3\] and the range is all real numbers. The relation is a function.

Step 1: Identifying Vertex and Direction

The parabola's vertex form is \(x=a(y-h)^{2}+k\), where \((h,k)\) is the vertex. In the given equation \(x=-4(y-1)^{2}+3\), the vertex is \((h,k)=(1, 3)\). The direction in which the parabola opens is determined by the leading coefficient. In this case, it is -4, which means the parabola opens to the left.

Step 2: Determining the Domain and Range

The domain of the function is the set of all possible x-values. Since the parabola opens left, the x values can be \(-\infty\) through 3 (including 3). So, the domain is \]\[-\infty, 3\]\. The range of the function is the set of all possible y-values. Since the parabola is vertical in orientation, it will include all y values, thus the range is \(-\infty, +\infty\) or all real numbers.

Step 3: Is the Relation a Function?

A relation is a function if and only if for each input, there is exactly one output. For a vertical parabola, given any x, there may be two corresponding y values, thus, it is not a function. For a horizontal parabola, given any y, there is exactly one x. As our parabola here opens to the left (it's horizontal), hence, it is a function.

Key Concepts

Vertex Form

The vertex form of a parabola is a useful way to express a quadratic equation. It helps us easily identify the vertex and the direction in which the parabola opens. The general equation is given by \(x = a(y-h)^2 + k\), where:

• \((h, k)\) represents the vertex of the parabola.
• \(a\) indicates the direction and width of opening.

In our example, the equation \(x = -4(y-1)^2 + 3\) shows:

• The vertex is at \((1, 3)\).
• The negative value of \(a = -4\) indicates that the parabola opens to the left.

The vertex is a critical point that represents the maximum or minimum point of the parabola depending on its direction. This makes vertex form a powerful tool to easily understand the parabola's structure.

Domain and Range

Understanding domain and range is essential in describing a function's behavior. For a parabola expressed in terms of \(x = a(y-h)^2 + k\), determining these sets is straightforward.
The **domain** is the collection of all possible \(x\)-values. In our equation, because the parabola opens to the left:

• The domain extends from \(-\infty\) to the vertex \(k\).
• Thus, the domain is \((-\infty, 3]\).

The **range** reflects all possible \(y\)-values. Given that the parabola can extend up and down forever:

• The range is all real numbers.

These concepts help us understand the extent and limits of the relation, offering insight into where the parabola "lives" on the coordinate plane.

Function Determination

Determining if a parabola is a function involves examining its orientation. A function has only one output (or one \(x\)) for each input (or \(y\)).
In the case of horizontal parabolas like \(x = a(y-h)^2 + k\):

• Each \(y\)-value corresponds to exactly one \(x\)-value.
• This aligns with the definition of a function.

For our equation, which opens to the left, this means it is indeed a function. Thid understanding helps us classify and predict the behavior of the relationship described by the equation. It's crucial for graphing and mathematical analysis, ensuring clarity in problem-solving.