Problem 44 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The solutions are approximately at the intersection points of the plots of a hyperbola and a parabola on a graph.

Step 1: Understand the Equations

We are given a system of equations: \( x^2 - y^2 = 3 \) and \( y = x^2 - 2x - 8 \). The first equation is a hyperbola, and the second is a parabola. We will plot both graphs to find their intersections.

Step 2: Plot the Parabola

The second equation \( y = x^2 - 2x - 8 \) is a parabola. To plot it, calculate values of \( y \) for several values of \( x \). For example, when \( x = 0 \), \( y = -8 \); when \( x = 2 \), \( y = -4 \). Connect these points to form a parabola that opens upwards.

Step 3: Plot the Hyperbola

The first equation can be rewritten as \( y^2 = x^2 - 3 \). This is a hyperbola centered at the origin. To plot it, find solutions for \( y \) for corresponding \( x \) values. Calculate points like \( (0, \sqrt{3}) \) and \( (0, -\sqrt{3}) \) and plot.

Step 4: Graphical Intersection

Overlay the plots of the hyperbola and the parabola on the same set of axes. The intersections of these graphs are the solutions to the system of equations. The intersection points can be read off the graph.

Step 5: Estimate Intersection Points

Upon examining the graph, estimate the coordinates of the intersection points. They represent the solutions to the system. Adjust the graph as necessary to ensure accuracy, and read the intersection points to two decimal places.

Key Concepts

System of Equations

A system of equations involves finding values for several variables that satisfy all given equations simultaneously. In our exercise, the system consists of two equations: a hyperbola and a parabola. When solving such problems graphically, we plot each equation on the same set of axes. New lines (
) are commonly introduced when students plot equations by picking convenient values that reveal the curve's nature visually. The points where their graphs intersect are the solutions we seek.

• Understanding why intersections matter: If two graphs meet at a point, the coordinates (x, y) satisfy both equations at once.
• The graphical method provides a visual verification of solutions, unlike algebraic methods which only offer numerical answers.
• To ensure accuracy, adjust your graph as necessary to improve the readability of intersection points.

Parabola

In our system, the second equation represents a parabola of the form \( y = x^2 - 2x - 8 \). A parabola is a symmetric curve that opens either upwards or downwards. Here, the parabola opens upwards because the coefficient of \( x^2 \) is positive.

• The vertex form is an excellent way to glean deeper insight. By completing the square, you can rewrite the equation to find the vertex of the parabola, which is its turning point.
• The axis of symmetry is a vertical line through the vertex. For our equation, the vertex formula \( x = -\frac{b}{2a} \) leads to \( x = 1 \).

This helps in plotting, as even minimal calculations reveal more about the graph's width and maximum or minimum points.

Hyperbola

The first equation \( x^2 - y^2 = 3 \) defines a hyperbola. Recognizable by its two separate curves (branches) that open sideways, a hyperbola's standard form is evident when rewritten as \( y^2 = x^2 - 3 \).

• Each branch approaches asymptotes, which are imaginary lines that guide the curve without touching it.
• Hyperbolas can open horizontally or vertically; here, the symmetry due to the origin suggests a standard shape familiar to those studying conics.

Plotting involves similar point calculations as with circles, except with hyperbolic properties in mind. Finding critical points like \( (0, \pm\sqrt{3}) \) reveals parts of the curve, from which the remaining graph can be extrapolated. An understanding of these fundamentals will help in correctly plotting the graph and finding intersections with the parabola.