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 (2.80, -1.36) and (-1.26, 0.06).

Step 1: Understand the System of Equations

The system of equations consists of two equations: \(x^2 - y^2 = 3\) and \(y = x^2 - 2x - 8\). We need to find points \((x, y)\) that satisfy both.

Step 2: Express the First Equation in Terms of y

For the first equation, \(x^2 - y^2 = 3\), express it in the form of a hyperbola: \(y^2 = x^2 - 3\). So, \(y = \pm\sqrt{x^2 - 3}\).

Step 3: Plot the Hyperbola

Plot the function \(y = \sqrt{x^2 - 3}\) and \(y = -\sqrt{x^2 - 3}\) on a graph. This represents the hyperbola, which is symmetric about the x-axis.

Step 4: Plot the Parabola

The second equation \(y = x^2 - 2x - 8\) is a parabola. Plot this parabola on the same graph as the hyperbola. This parabola opens upwards.

Step 5: Find Intersection Points

Look for points where the parabola intersects with the hyperbola. These points are solutions to the system of equations. Solutions occur where the graphs meet.

Step 6: Approximate Intersection Points

Through graphical estimation, determine the approximate intersection points. Calculate their values and round them to two decimal places. Possible points are around \((2.80, -1.36)\) and \((-1.26, 0.06)\).

Step 7: Verify Solutions

Check if the estimated intersection points satisfy both original equations to verify accuracy. Use substitution and arithmetic to confirm.

Key Concepts

Graphical Method

When solving systems of equations, the graphical method can be incredibly intuitive. It involves plotting each equation on the same graph to visually find their points of intersection. Each of these points, where the graphs meet, represents a solution to the system. This technique is particularly helpful because:

• It provides a visual representation of the solutions.
• It helps to understand the relationship between equations.
• It makes it easier to estimate solutions when exact ones are difficult to calculate.

To use the graphical method effectively, you'll need to:
- Understand what each equation represents (a line, parabola, circle, etc.).
- Accurately plot each equation on a coordinate grid.
- Look for points of intersection as potential solutions.
Remember, this method may require estimation, especially if the intersection points do not fall on grid lines exactly. It is often used together with algebraic methods to verify solutions.

Hyperbola

A hyperbola is a type of curve on a graph and is similar to a set of opposing, open-ended u-shaped curves. It can often resemble two mirrored curves extending towards infinity. In our problem, the first equation, \(x^2 - y^2 = 3\), can be rewritten as \(y^2 = x^2 - 3\) or \(y = \pm\sqrt{x^2 - 3}\). This shows the two branches of the hyperbola,
with each branch corresponding to one equation: \(y = \sqrt{x^2 - 3}\) and \(y = -\sqrt{x^2 - 3}\).

The properties of a hyperbola:

• It has two distinct branches that are mirror images across both axes or one of them.
• It is symmetrical relative to its axes.
• Has asymptotes, which are lines that the hyperbola approaches but never touches.

Understanding these characteristics can help you accurately plot the hyperbola and identify its intersections with other curves like parabolas.

Parabola

A parabola is a symmetrical, open plane curve formed by a quadratic equation. In this exercise, the parabola is given by the equation \(y = x^2 - 2x - 8\). It has some defining features which make it recognizable:

• It is symmetrical about its vertex, which is the highest or lowest point depending on the direction it opens.
• It might open upwards or downwards based on the sign of the leading coefficient in the quadratic equation (\(x^2\) term).
• The vertex form of a parabola makes it easy to find its vertex, and this form can be derived by completing the square.

For the equation \(y = x^2 - 2x - 8\), you can determine its vertex by using the formula \(x = \frac{-b}{2a}\), where \(a\) is the coefficient of \(x^2\), and \(b\) is the coefficient of \(x\).
Recognizing this will help when plotting the parabola and checking where it intersects with the hyperbola.