Problem 69 - College Algebra

Question

Which one of the following is true? a. The parabola whose equation is \(x=2 y-y^{2}+5\) opens to the right. b. If the parabola whose equation is \(x=a y^{2}+b y+c\) has its vertex at \((3,2)\) and \(a>0,\) then it has no \(y\) -intercepts. c. Some parabolas that open to the right have equations that define \(y\) as a function of \(x .\) d. The graph of \(x=a(y-k)+h\) is a parabola with vertex at \((h, k)\)

Step-by-Step Solution

Answer

The statement \(a. The parabola whose equation is \(x=2y - y^2 + 5\) opens to the right\) is true.

Step 1: Analyzing the first statement

The general equation of the parabola \(x = ay^2 + by + c\) opens to right if a < 0 and opens to the left if a > 0. The given equation in the first statement is \(x=2y - y^2 + 5\). Here the coefficient of \(y^2\) is -1, which is less than 0. Thus, the parabola opens to the right. So, the first statement is correct.

Step 2: Analyzing the second statement

The vertex of the parabola \(x=ay^2+by+c\) is given by \(-b/2a\). If (3, 2) is the vertex, then 2 equals \(-b/2a\), leading to \(b = -4a\). For \(a > 0\), this implies \(b < 0\). However, this won't necessarily restrict \(y\)-intercepts as to have no \(y\)-intercepts c should also be greater than zero also which is not available, Hence, the second statement is not necessarily true.

Step 3: Analyzing the third statement

The general equation of a parabola can be written in two forms: either \(x\) expressed as a function of \(y\) (i.e., \(x = f(y)\)) or \(y\) expressed as a function of \(x\) (i.e., \(y = f(x)\)). The form \(y = f(x)\) describes parabolas that open up or down, whereas the form \(x = f(y)\) describes parabolas that open left or right. So, it's possible for a parabola opening to the right to have its equation define \(y\) as a function of \(x\). Therefore, the third statement is false.

Step 4: Analyzing the fourth statement

The equation \(x = a(y - k) + h\) is not the standard form of a parabola. In general, the standard form of a parabola is either \(x = a(y - k)^2 + h\) or \(y = a(x - h)^2 + k\). Therefore, the fourth statement is false.

Key Concepts

Parabola Orientation

The orientation of a parabola is determined by the direction it opens. In general, the equation of a parabola can be expressed as either opening horizontally or vertically. The form of the equation plays a crucial role in determining its orientation and is typically defined as:

• For parabolas opening upwards or downwards, the equation is usually written in the form: \( y = ax^2 + bx + c \). Here, if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• For parabolas opening left or right, the equation is typically: \( x = ay^2 + by + c \). In this case, if \( a > 0 \), the parabola opens to the left, and if \( a < 0 \), it opens to the right.

Understanding the orientation is vital because it influences how the parabola is represented in graphs. In the exercise's first statement, the equation \( x = 2y - y^2 + 5 \) has a coefficient of \(-1\) for \( y^2 \), indicating the parabola opens to the right.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. It is a critical feature since it represents the highest or lowest point on the graph, depending on the parabola's orientation. For parabolas in the form of \( y = ax^2 + bx + c \) or \( x = ay^2 + by + c \), determining the vertex involves:

• For vertical parabolas, the vertex can be found using the formula \( x = -\frac{b}{2a} \).
• For horizontal parabolas, \( y = -\frac{b}{2a} \) gives the vertex.

In the context of the exercise, the vertex of the parabola serves as a reference point. If the vertex is given, as in the case "vertex at (3,2)," you can use it to determine other properties or confirm calculations in your problems. Remember, the vertex is sometimes not enough to determine intercepts, as seen in statement b of the exercise, where having \( a > 0 \) doesn't guarantee no \( y \)-intercepts.

Standard Form of a Parabola

The standard form of a parabola simplifies analysis and graphing by highlighting its key features, like vertex and direction. There are mainly two standard equations for parabolas:

• For vertical parabolas, the equation is \( y = a(x - h)^2 + k \). The parameters \( h \) and \( k \) are the coordinates of the vertex.
• For horizontal parabolas, the form is \( x = a(y - k)^2 + h \). Similarly, \( h \) and \( k \) denote the vertex's place.

This form is beneficial because it makes it straightforward to identify the vertex directly from the equation. For instance, the incorrect form given in statement d, \( x = a(y-k)+h \), does not mirror the typical structure and thus fails to deliver the correct parabola features. The presence of squared terms is essential to maintain the equation's parabolic nature.

Function Definition of Parabola

A parabola can either define a function or not, depending on its orientation. When a parabola opens upwards or downwards, it can be expressed as a function of \( x \) because for each \( x \), there is exactly one \( y \). However, when parabolas open left or right, they do not define a function of \( y \) by typical standards because for each \( y \) value, there may be more than one \( x \). To be a function, every input should map to a single output.In statement c of the exercise, it is pointed out that some parabolas opening to the right cannot define a function of \( y \) due to the multiple \( x \) values possibility. The vertical line test is a simple approach to check if an equation defines a function. If any vertical line crosses the graph at more than one point, it does not represent a function in its conventional sense.