Problem 51 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x^{2}+y^{2} &=8 \\ y &=x^{2} \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to this system of equations are the intersection points of the two graphs, as identified and verified with a graphing utility.

Step 1: Graph the equations

Plot the equations \(x^{2}+y^{2} =8\) and \(y =x^{2}\) respectively using any graphing tool. The points where the graphs intersect are the solutions to the given system of equations.

Step 2: Find intersection points

From the graph, identify and record all the points of intersection. Each point will be of the form (x, y). Make sure to approximate to three decimal places.

Step 3: Verify solutions

Substitute the recorded values into the original system. For each point (x, y), x² + y² should be equal to 8 and y should be equal to x². This verifies that the points are indeed the solution to the system.

Key Concepts

Intersection Points

In any system of equations, intersection points are crucial as they represent the solutions where the equations meet. If we consider the system given, it contains two equations: a circle and a parabola. The circle's equation is \(x^2 + y^2 = 8\). The parabola's equation is \(y = x^2\).

To find the intersection points:

• Graph both equations on the same coordinate plane.
• Identify where the lines of the circle and parabola cross each other.

Each intersection in this context corresponds to a particular set of \((x, y)\) values that satisfy both equations simultaneously. These intersection points can be seen clearly using a graphing calculator or software that allows for precise plotting. For this exercise, round each value to three decimal places to ensure accuracy in your approximate solutions.

Graphing Utility

A graphing utility is an invaluable tool when dealing with complex systems of equations, especially those involving curves like circles and parabolas. It allows you to visually understand where these shapes intersect without having to rely on solving equations algebraically, which can often be more challenging.

Benefits of using a graphing utility include:

• It provides a clear visual representation of the problem at hand.
• It simplifies the process of identifying intersection points.
• It allows for easy adjustments and experimentation by changing values or scales.

Choose a graphing tool—whether it's a handheld graphing calculator or an online graphing application—and input the equations \(x^2 + y^2 = 8\) and \(y = x^2\). The points where these graphs intersect are the solutions you're seeking.

Quadratic Equations

A quadratic equation is typically in the form \(ax^2 + bx + c = 0\). These equations can represent various geometrical curves, commonly parabolas. In the current exercise, one of the equations \(y = x^2\) is a simple quadratic with the shape of a standard upward-facing parabola.

Key features of this quadratic equation are:

• The vertex at the origin \((0, 0)\) in this case, since there are no additional terms shifting it.
• An axis of symmetry along the y-axis.
• Intersecting a circle given by \(x^2 + y^2 = 8\) can provide multiple solutions—both positive and negative directions relative to the x-axis.

Understanding the behavioral characteristics of quadratic equations helps predict where to expect intersection points when overlaid with other shapes like circles.

Verification of Solutions

After locating the intersection points with a graphing utility, verifying these solutions ensures they are correct. Verification involves substituting the intersection point values back into the original system of equations and checking for consistency.

Steps to verify solutions:

• Take each intersection point \((x, y)\) and plug it into the equation \(x^2 + y^2 = 8\). Confirm that this equality holds true; this checks the point's validity for the circle.
• Then, substitute \(x\) into \(y = x^2\) to confirm that the \(y\) value matches; this verifies the point for the parabola.

This process ensures the solutions are accurate and that both equations in the system are satisfied. By thoroughly verifying, one gains confidence in the correctness of the approximated solutions.