Problem 43 - Precalculus: Mathematical for Calculus

Question

Use the graphical method to find all solutions of the system of equations, correct to two decimal places. $$\left\\{\begin{array}{l}\frac{x^{2}}{9}+\frac{y^{2}}{18}=1 \\\y=-x^{2}+6 x-2\end{array}\right.$$

Step-by-Step Solution

Answer

The solutions are approximately (1.21, 5.34) and (4.79, 4.74).

Step 1: Understand the System of Equations

We have two equations:1. \( \frac{x^{2}}{9} + \frac{y^{2}}{18} = 1 \), which is an equation of an ellipse.2. \( y = -x^{2} + 6x - 2 \), which is a quadratic equation forming a parabola. Our goal is to find the points where these two graphs intersect.

Step 2: Graph the Ellipse

The equation \( \frac{x^{2}}{9} + \frac{y^{2}}{18} = 1 \) can be written as \( x^2/3^2 + y^2/(3\sqrt{2})^2 = 1 \). This tells us the ellipse is centered at the origin (0,0) with semi-major axis \( 3\sqrt{2} \) along the y-axis and semi-minor axis 3 along the x-axis. Plot this ellipse on a coordinate plane.

Step 3: Graph the Parabola

The equation \( y = -x^2 + 6x - 2 \) is a downward opening parabola. Identify the vertex and intercepts. The vertex can be found using \( x = -\frac{b}{2a} = 3 \), giving \( y = 7 \). Plot the vertex at (3,7) and find additional points by choosing x values. Plot this curve on the same graph used for the ellipse.

Step 4: Find Intersections

Check for points where the graphs intersect by plotting carefully. These points represent the solutions to the system. You'll realize the curves intersect at two points, which we estimate visually and can verify through substitution or using graphing tools.

Step 5: Refine Solutions to Two Decimal Places

From the graph or computation tool, you'll find the points of intersection of the ellipse and parabola at approximately (1.21, 5.34) and (4.79, 4.74). These values should be rounded to two decimal places as required.

Key Concepts

Ellipse

An ellipse is a fascinating geometric shape that resembles a stretched circle. It forms when you slice through a cone at an angle. The standard equation for an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. The larger value of \(a\) or \(b\) indicates the direction in which the ellipse is stretched. In our case, the equation \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \) tells us that the ellipse is centered at the origin (0,0).

• The semi-major axis, \(3\sqrt{2}\), is along the y-axis, meaning the ellipse is taller.
• The semi-minor axis, 3, lies along the x-axis.

Visualize this ellipse as an orbit around the center, stretched more vertically. When graphing an ellipse, it's crucial to determine these axes accurately so that you can draw it correctly and see how other curves interact with it.

Parabola

A parabola is a U-shaped curve that appears in many natural and mathematical situations, such as the path of a thrown ball or the graph of a quadratic function. The general form of a parabolic equation is \( y = ax^2 + bx + c \). In the equation \( y = -x^2 + 6x - 2 \), we notice that the coefficient \(- 1\) before \(x^2\) makes this a downward opening parabola. The shape indicates that it has a maximum point at its vertex.

• To find the vertex, use the formula \( x = -\frac{b}{2a} \). For this equation, \( x = 3 \), and substituting back gives \( y = 7 \).
• This means the vertex of the parabola is at (3,7), a critical point you should plot first when sketching the graph.

When graphing, other features to check are the x and y intercepts, which help complete the picture. Parabolas can intersect with other shapes, and recognizing their paths helps in solving system equations like the one here.

System of Equations

A system of equations consists of two or more equations with common variables. The goal is to find the set of values for these variables that satisfy all equations simultaneously.In this exercise, we have an ellipse and a parabola forming the system:

• \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \)
• \( y = -x^2 + 6x - 2 \)

The graphical method involves plotting both equations on the same coordinate plane to visually identify their intersection points, which represent the solutions. These solutions are where both equations are true at the same time.

• Intersections are checked by carefully observing where one curve crosses another.
• Finding precise intersections might require substitution or computational tools if the graph is not sufficient on its own.

By graphing these equations, you determine the potential solution points at approximately (1.21, 5.34) and (4.79, 4.74), which should be rounded to the required precision. Understanding the graphical methods enhances problem-solving intuition for these types of equations.