Problem 79 - College Algebra with Modeling and Visualization

Question

Graph the parabola. $$x=2.3(y+1)^{2}$$

Step-by-Step Solution

Answer

The parabola opens to the right with vertex at (0, -1).

Step 1: Identify the Form of the Equation

The given equation is \( x = 2.3(y+1)^2 \). This is a quadratic equation in the form of \( x = a(y - k)^2 + h \), where \( a = 2.3 \), \( h = 0 \), and \( k = -1 \). This indicates a parabola that opens to the right.

Step 2: Determine the Vertex of the Parabola

The vertex of a parabola in the form \( x = a(y - k)^2 + h \) is \((h, k)\). Here, \( h = 0 \) and \( k = -1 \), so the vertex is at the point \((0, -1)\).

Step 3: Identify the Direction of Opening

Since the equation is \( x = 2.3(y+1)^2 \) and the coefficient \( a = 2.3 \) is positive, the parabola opens to the right. In this form, positive \( a \) values indicate a parabola that opens to the right, and negative \( a \) values indicate it opening to the left.

Step 4: Find Additional Points for Graphing

To graph the parabola accurately, choose some values for \( y \) and solve for \( x \). For example:- If \( y = 0 \): \( x = 2.3(0+1)^2 = 2.3 \).- If \( y = 1 \): \( x = 2.3(1+1)^2 = 9.2 \).- If \( y = -2 \): \( x = 2.3(-1)^2 = 2.3 \).These points are (2.3, 0), (9.2, 1), and (2.3, -2).

Step 5: Sketch the Graph

Plot the vertex (0, -1) and the additional points (2.3, 0), (9.2, 1), and (2.3, -2) on a coordinate plane. Draw a smooth curve through these points ensuring that it opens to the right. The curve should be symmetric with respect to the horizontal line \( y = -1 \).

Key Concepts

Understanding the Parabola

A parabola is a U-shaped curve that you may encounter when dealing with quadratic equations. In this case, the equation takes the form \( x = 2.3(y + 1)^2 \). This represents a special type of parabola aligned with the y-axis, which means the opening direction is along the x-axis rather than the more common y-axis alignment.

• For standard parabolas, the vertex is the central point, and the curve is symmetrical around it.
• The direction in which a parabola opens is determined by the sign of the coefficient in front of the squared term.

In our example, because the equation is in the form \( x = a(y - k)^2 + h \), and since \( a \) (2.3) is positive, it indicates our parabola opens to the right. The vertex, an essential characteristic of any parabola, is located at the point \((0, -1)\), marking the lowest or highest spot on a vertically oriented parabola or the leftmost for a horizontally oriented one.

Exploring the Vertex Form

The vertex form of a parabola provides a convenient way to determine critical attributes like the vertex and direction. The vertex form can be written as \( x = a(y - k)^2 + h \), resembling a mirror image of the typical \( y = a(x - h)^2 + k \) form used for parabolas opening along the y-axis.

• The vertex form directly gives us the vertex of the parabola at the point \((h, k)\).
• The sign and value of the coefficient \( a \) dictate whether the parabola opens to the left or right.

Since our equation is \( x = 2.3(y + 1)^2 \), it fits perfectly into the vertex form structure with \( h = 0 \) and \( k = -1 \). This tells us the parabola's vertex is at \((0, -1)\), providing a starting point for sketching or analyzing the curve.

Graphing the Parabola

Graphing a parabola involves plotting its key points and understanding its general shape. Here, you would start by plotting the vertex, \((0, -1)\), since this represents the central point of symmetry for your parabola.

• Start by plotting the vertex to provide a clear center point for symmetry.
• Choose a few values of \( y \) to find additional points along the parabola.
• For accurate plotting, calculate \( x \) for each chosen \( y \) value.

In our example, by picking some easy-to-calculate points such as \( y = 0 \), \( y = 1 \), and \( y = -2 \), we found points \((2.3, 0)\), \((9.2, 1)\), and \((2.3, -2)\). Plotting these alongside the vertex helps outline the parabola's rightward curve. Once you have plotted these points, draw a smooth, symmetric curve through them to sketch the graph accurately. The parabola is symmetrical about the line \( y = -1 \), ensuring both sides mirror each other perfectly around the vertex.