Problem 38 - Precalculus Mathematics for Calculus

Question

Use the graphical method to find all solutions of the system of equations, rounded to two decimal places. $$\left\\{\begin{array}{l} x^{2}-y^{2}=3 \\ y=x^{2}-2 x-8 \end{array}\right.$$

Step-by-Step Solution

Answer

The solutions are approximately (4.08, 2.64) and (-2.08, -2.16).

Step 1: Understand the Given Equations

The first equation is a hyperbola: \(x^2 - y^2 = 3\). The second is a parabola: \(y = x^2 - 2x - 8\). We will graph both to find their points of intersection.

Step 2: Graph the Hyperbola

Rearrange the first equation to express \(y\) in terms of \(x\): \(y^2 = x^2 - 3\). Solving for \(y\) gives \(y = \pm\sqrt{x^2 - 3}\). Graph both branches (the positive and negative) by plotting the hyperbola centered at the origin and intersecting the x-axis at \(x = \pm \sqrt{3}\).

Step 3: Graph the Parabola

The equation \(y = x^2 - 2x - 8\) is a standard parabola facing upwards (since the coefficient of \(x^2\) is positive). Find its vertex by completing the square: \(y = (x-1)^2 - 9\). Thus, the vertex is at \((1, -9)\). Plot this parabola.

Step 4: Identify Points of Intersection

On the graph, observe where the hyperbola and parabola intersect. These points represent the solutions to the system. You might initially find these points approximately through estimation or using graph plotting software.

Step 5: Verify and Round the Solutions

Plug the estimated intersection points back into both original equations to verify. Round the coordinates to two decimal places, ensuring consistent intersection points are solutions to both equations.

Key Concepts

Hyperbola

A hyperbola is a type of conic section that can be defined by the equation \[x^2 - y^2 = 3\]. In this equation, the curve represents two mirrored arcs, creating two separate curves that diverge away from each other. The equation reveals that this hyperbola is centered at the origin, because there are no additional constants added to the squared terms. It intersects the x-axis at \(x = \pm\sqrt{3}\).
Key features of a hyperbola include:

• Asymptotes: These are lines the hyperbola approaches but never actually touches. Our equation’s asymptotes would be the lines \(y = x\) and \(y = -x\) for a basic configuration.
• Branches: Each hyperbola has two separate curves known as branches. They often mirror each other across the origin or another central point.

Understanding how to graph and interpret these features is crucial for finding intersection points with other graphs, such as parabolas.

Parabola

A parabola is another kind of conic section, often appearing as a U-shaped curve on a graph. The general form for a parabolic equation is \(y = ax^2 + bx + c\), which describes how it opens either upwards or downwards. In our problem, we work with \(y = x^2 - 2x - 8\).
Here's what you need to know:

• Vertex: The highest or lowest point on a parabola. For \(y = x^2 - 2x - 8\),the vertex is found by completing the square, yielding \((1, -9)\).
• Direction: A parabola faces up if the quadratic term (\(x^2\) in this case) is positive, and down if negative. This curve opens upwards.
• Axis of Symmetry: This is the vertical line that splits the parabola into two symmetrical halves. It's equation here is \(x = 1\).

Parabolas often intersect with other curves, making them crucial for solving systems graphically.

Intersection Points

Intersection points between two curves are the coordinates where their graphs meet. In our exercise, they are the solutions to the system of equations involving both the hyperbola and the parabola. To find these intersection points:
Graphically superimpose the hyperbola and parabola on the same coordinate plane. Pay attention to the curves' paths to accurately identify where they cross each other. These intersection points are found where:

• The branch of the hyperbola intersects with the curve of the parabola.
• Each point satisfies both equations simultaneously.

Checking the estimated coordinates is essential. Substitute these solutions back into the original equations to ensure they solve each equation. Remember to round the values to two decimal places, according to the precision required by the task. This graphical method of finding intersection points helps visualize and verify the solutions more effectively.