Problem 62 - 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? $$ x=-3(y-1)^{2}-2 $$

Step-by-Step Solution

Answer

The parabola with the equation \( x = -3(y-1)^{2} - 2 \) opens to the left with a vertex at the point (-2, 1). Hence, its domain is \((-\infty , -2]\) and the range is \((-\infty , \infty)\). Since any vertical line crosses the parabola at most at one point, the relation is a function.

Step 1: Determine the Orientation and Vertex of the Parabola

Given the equation \( x = -3(y-1)^{2} - 2 \), the vertex form of a parabolic equation is \[ x = a(y - h)^{2} + k \] where (k, h) is the vertex of the parabola. This is a vertical parabola that opens to the left because the coefficient of \((y - h)^2\) is negative. The vertex of the parabola is at the point (-2, 1).

Step 2: Determine the Domain and Range

Since the parabola opens to the left, all the y-values (up and down) will be covered, but the x-values will only be from -Infinity to the vertex x-value -2. Therefore, the domain is \((-\infty , -2]\) and the range is \((-\infty , \infty)\).

Step 3: Determine if the Relation is a Function

To determine whether the relation is a function, apply the vertical line test. If any vertical line drawn through the parabola crosses at more than one point, then the relation is not a function. Since this parabola opens to the left, any vertical line would cross the parabola at most at one point. Therefore, the relation is a function.

Key Concepts

Vertex Form of a Parabola

Understanding the vertex form of a parabola is essential in analyzing its characteristics and graphing the parabolic curve. The vertex form is particularly useful because it provides immediate details about the parabola's vertex — the highest or lowest point on the curve, depending on its orientation. The general form of a parabola in vertex form is
\[ y = a(x - h)^2 + k \]
or
\[ x = a(y - k)^2 + h \]
depending on whether it's horizontally or vertically oriented. In these equations, \( (h, k) \) represents the coordinates of the vertex, and the value of \( a \) affects the width and direction of the opening of the parabola. A positive value of \( a \) indicates the parabola opens upwards (for a y-equation) or to the right (for an x-equation), while a negative \( a \) flips the parabola downwards or to the left respectively. By analyzing the given equation
\[ x = -3(y - 1)^2 - 2 \],
we see that the vertex is at \( (-2, 1) \), indicating that the curve reaches its furthest point on the x-axis at -2 and on the y-axis at 1.

Parabolic Orientation

The orientation of a parabola is a significant feature that affects its domain and range. Orientation refers to the direction in which a parabola opens. There are two primary orientations for a parabola: vertically or horizontally. A vertical parabola, represented by the form \( y = a(x - h)^2 + k \), can open upwards if \( a > 0 \) or downwards if \( a < 0 \). Conversely, a horizontal parabola, given by the form \( x = a(y - k)^2 + h \), opens to the right if \( a > 0 \) or to the left if \( a < 0 \).

Impact on Domain and Range

For a vertical parabola, the domain is all real numbers, and the range is limited by the vertex. However, for a horizontal parabola, like our exercise example, the orientation changes everything: the domain is capped by the vertex's x-value, while the range encompasses all real numbers because it can extend infinitely up and down.

Vertical Line Test

The vertical line test is a visual way to determine if a curve on a graph represents a function. It's based on the definition of a function, which states that for each input value, there should be exactly one output value. To apply the vertical line test, imagine drawing vertical lines through various points along the x-axis. If any of these lines intersect the graph in more than one place, the graph does not pass the test, and the curve does not represent a function.
For our example parabola denoted by
\[ x = -3(y - 1)^2 - 2 \],
the parabola opens to the left. A vertical line drawn at any point on the graph will only touch the curve once, indicating that for every x-value, there is only one corresponding y-value. Therefore, the given relation passes the vertical line test and is indeed a function. This is a crucial concept in mathematics because it helps to verify the behavior of equations when graphed and solidify understanding of functions versus non-functions.