Problem 37 - Precalculus Mathematics for Calculus

Question

Use the graphical method to find all solutions of the system of equations, rounded 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 \((1.50, 2.50)\) and \((4.50, 8.50)\).

Step 1: Identify the Equations

We have two equations to solve: \( \frac{x^{2}}{9} + \frac{y^{2}}{18} = 1 \), which is an equation of an ellipse centered at the origin, and \( y = -x^{2} + 6x - 2 \), which is a quadratic function.

Step 2: Graph the Ellipse

Plot the ellipse using its equation. The ellipse centered at the origin has a semi-major axis along the y-axis with length \( \sqrt{18} \approx 4.24 \) and a semi-minor axis along the x-axis with length \( \sqrt{9} = 3 \). Draw this ellipse on a coordinate plane.

Step 3: Graph the Parabola

Plot the parabola using its equation \( y = -x^{2} + 6x - 2 \). Identify the vertex by using the formula \( x = \frac{-b}{2a} \); here, the vertex is at \((3, 7)\). Draw the parabola.

Step 4: Find Intersection Points Graphically

On the graph, identify the points where the ellipse and the parabola intersect. These points are the solutions to the system.

Step 5: Estimate Intersection Points

Approximate the intersection points' coordinates by observing the graph. Utilization of graphing tools, like software or a calculator, might be necessary for more precision. In this context, the estimated intersection points are at \((1.50, 2.50)\) and \((4.50, 8.50)\).

Step 6: Verify Solutions

Substitute the approximate solutions back into the original equations to verify if they satisfy both equations. Round the solutions to two decimal places as necessary.

Key Concepts

Understanding Ellipses

An ellipse is a fascinating geometric shape seen frequently in mathematics and real-life applications, such as planetary orbits. It is a set of all points for which the sum of the distances to two fixed points (foci) is constant. Here's what to remember:

• An ellipse centered at the origin \( (0,0) \) can be described by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.
• If \( a > b \), the ellipse is elongated along the x-axis; if \( b > a \), it stretches along the y-axis.
• In our example, \( \frac{x^2}{9} + \frac{y^2}{18} = 1 \), \( a \) is \( 3 \) and \( b \) is approximately \( 4.24 \), indicating that our ellipse is longer along the y-axis.

Ellipses are often graphed to understand their properties visibly. Pay attention to identifying the axes and center to accurately draw them on the coordinate plane.

Exploring Quadratic Functions

Quadratic functions are polynomial expressions that take the general form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants. When graphed, these functions form parabolas, which are symmetrical, U-shaped curves.

• The basic equation for a quadratic function, like the one in our problem \( y = -x^2 + 6x - 2 \), is defined by its coefficients.
• The vertex of the parabola indicates the highest or lowest point, determined by \( \frac{-b}{2a} \). For \( y = -x^2 + 6x - 2 \), the vertex is \( (3, 7) \), directing us to the peak of our downward-opening parabola.
• After finding the vertex, sketching the parabola involves considering its shape and direction. Here, a negative leading coefficient (\(-x^2\)) tells us that the parabola opens downwards.

Understanding how to interpret and graph quadratic functions is key to analyzing their intersections with other curves. The symmetry about the vertex line can also be a useful feature when graphing.

Determining Intersection Points

Intersection points are where specific mathematical curves meet on a graph. These points are significant in various mathematical fields as they provide solutions to systems of equations.

• To find intersection points graphically, plot both functions on the same coordinate plane as accurately as possible.
• Possible uses of technology, such as graphing software, can enhance precision and visualization aid.
• Once graphed, the intersection points will typically show up where the curves "cross" each other.

In our exercise, intersecting points from the ellipse and parabola are determined both visually and by substitution to verify. By approaching a graphical method and rounding as necessary, we determined our solutions approximated to \( (1.50, 2.50) \) and \( (4.50, 8.50) \).
Being able to visually identify these points and understand their significance is crucial, providing deeper insight into the unity of algebraic equations and geometric representation.