Problem 63 - A Graphical Approach to College Algebra

Question

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}-y=0\\\&x^{2}+y^{2}=2\end{aligned}$$

Step-by-Step Solution

Answer

The solutions are (1, 1) and (-1, 1).

Step 1: Graph the Equation x^2 - y = 0

First, rearrange the equation \(x^2 - y = 0\) to solve for \(y\):\[y = x^2\]. This is a parabola that opens upwards. Plot this graph by finding points such as \((0, 0)\), \((1, 1)\), \((-1, 1)\), \((2, 4)\), and \((-2, 4)\). Connect these points smoothly to complete the parabola.

Step 2: Graph the Equation x^2 + y^2 = 2

This equation describes a circle centered at the origin with radius \(\sqrt{2}\). To graph this, find and plot intercepts such as \((\sqrt{2}, 0)\), \((0, \sqrt{2})\), \((-\sqrt{2}, 0)\), and \((0, -\sqrt{2})\). Connect these points to form a circle around the origin.

Step 3: Identify Points of Intersection

Look for points where the parabola from Step 1 intersects with the circle from Step 2. These points are visually where the graphs overlap.

Step 4: Check Solutions Algebraically

To verify the intersection points, solve the system algebraically. Substitute \(y = x^2\) into \(x^2 + y^2 = 2\) to get \(x^2 + (x^2)^2 = 2\). Simplify to \(x^2 + x^4 = 2\), then \(x^4 + x^2 - 2 = 0\). Factor to find \((x^2 - 1)(x^2 + 2) = 0\). Solving gives \(x^2 = 1\) (since \(x^2 + 2 = 0\) has no real solutions), so \(x = \pm 1\). Substitute back to find \(y\) as \(1\). The solutions are \((1, 1)\) and \((-1, 1)\).

Step 5: Verify Graphically

Check that the points \((1, 1)\) and \((-1, 1)\) are indeed where the graphs intersect on your sketch. Both points satisfy \(y = x^2\) and lie on the circle \(x^2 + y^2 = 2\), confirming their correctness.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of unknowns. Solving them means finding the values of the unknowns that satisfy all the equations simultaneously. In this exercise, we look at graphical solutions where two graphs representing the equations are plotted on the same set of axes and intersections reveal possible solutions.

• Equation 1: \(x^2 - y = 0\), simplifies to \(y = x^2\)
• Equation 2: \(x^2 + y^2 = 2\)

By plotting these, we easily see where they overlap, indicating where both conditions are met. Graphical solutions are intuitive and visually represent how different mathematical entities interact. They are useful in confirming solutions and understanding the relationship between variables.
Notes: It's essential to ensure the accuracy of the plotted graphs for correct intersection points.

Parabola

A parabola is a U-shaped curve that is symmetrical. It is the graph of a quadratic equation, such as \(y = x^2\). In this vertex form, the parabola opens upwards, and its vertex is at the origin.
Key points defining a parabola help in plotting it:

• The vertex, which is the highest or lowest point depending on direction. Here it's \((0, 0)\).
• Axis of symmetry, a vertical line that splits the parabola into two mirrored halves.
• Standard points, like \((1, 1)\) and \((-1, 1)\), which help in shaping the curve.

Understanding the nature of a parabola, including its vertex and line of symmetry, is crucial for accurate graphing. It reflects how the quadratic relationship creates the characteristic curve.

Circle

The circle in this exercise is described by the equation \(x^2 + y^2 = 2\). This represents a perfect circle centered at the origin with a radius of \(\sqrt{2}\).
Some features of a circle include:

• The center point, which coordinates both axes equally. Here, it’s \((0, 0)\).
• Radius, the constant distance from the center to any point on the circle. Here it's \(\sqrt{2}\).
• Symmetry, as a circle maintains consistency in all directions.

Graphing a circle involves plotting points equidistant from the center and smoothly connecting them. It reflects uniformity and symmetry in geometry. In context, the circle helps create intersections with other graphs that reveal solutions to the system of equations.