Describing The System Of Equations: A Detailed Guide

Hey guys! Let's dive into the fascinating world of systems of equations, specifically focusing on how to describe them accurately. In this article, we're going to break down the system:

$$\begin{array}{l} x^2 + y^2 = 2 \\ y = 2x^2 - 3 \end{array}$$

We will explore each equation individually and then discuss how they interact within the system. So, buckle up and let's get started!

Understanding the First Equation: x² + y² = 2

When we first encounter x² + y² = 2, it's crucial to recognize its form. This equation screams circles! More specifically, it represents a circle centered at the origin (0, 0). But how do we know this? Let's break it down.

The Circle Equation

The general equation of a circle with a center at (h, k) and a radius r is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

In our case, the equation is x² + y² = 2. Comparing this with the general form, we can see that h = 0, k = 0, and r² = 2. This tells us that the center of the circle is indeed at the origin (0, 0). Now, what about the radius?

Determining the Radius

Since r² = 2, we can find the radius by taking the square root of both sides:

$$r = \sqrt{2}$$

So, the radius of our circle is √2. This means the circle extends √2 units in all directions from the center (0, 0). To put this into perspective, √2 is approximately 1.414, so the circle is a bit larger than a circle with a radius of 1.

Visualizing the Circle

Imagine a circle perfectly centered at the intersection of the x and y axes. It stretches out about 1.414 units in each direction – left, right, up, and down. This visual representation is crucial for understanding how this equation behaves and how it interacts with other equations in the system.

Key Takeaways

• The equation x² + y² = 2 represents a circle.
• The circle's center is at the origin (0, 0).
• The radius of the circle is √2.

Now that we've thoroughly dissected the first equation, let's move on to the second equation and see what it brings to the table.

Analyzing the Second Equation: y = 2x² - 3

Okay, let's tackle the second equation in our system: y = 2x² - 3. This one looks quite different from the first, doesn’t it? Instead of a circle, we're dealing with a parabola. But what kind of parabola? Let's break it down step by step.

Recognizing the Parabola

The general form of a parabola that opens either upwards or downwards is:

$$y = ax^2 + bx + c$$

Comparing this to our equation, y = 2x² - 3, we can see that it perfectly fits this form. Here, a = 2, b = 0, and c = -3. The coefficient 'a' plays a crucial role in determining the parabola's shape and direction. Since a = 2 is positive, the parabola opens upwards. If 'a' were negative, it would open downwards.

Finding the Vertex

The vertex is the most crucial point on a parabola – it’s the turning point. For a parabola in the form y = ax² + bx + c, the x-coordinate of the vertex (h) can be found using the formula:

$$h = -\frac{b}{2a}$$

In our case, b = 0 and a = 2, so:

$$h = -\frac{0}{2 * 2} = 0$$

The x-coordinate of the vertex is 0. To find the y-coordinate (k), we substitute x = 0 back into the equation y = 2x² - 3:

$$y = 2(0)^2 - 3 = -3$$

Thus, the vertex of our parabola is at the point (0, -3). This is the lowest point on the parabola, since it opens upwards.

Understanding the Shape

The coefficient 'a' not only tells us the direction the parabola opens but also how “wide” or “narrow” it is. A larger absolute value of 'a' means the parabola is narrower, while a smaller value means it's wider. In our case, a = 2, which means the parabola is a bit narrower than the standard parabola y = x².

Visualizing the Parabola

Picture a U-shaped curve sitting on the coordinate plane. The very bottom of the U is at the point (0, -3). The curve opens upwards, becoming steeper as you move away from the vertex in either direction. This mental image is key to understanding the parabola's behavior and its interactions with the circle.

Key Takeaways

• The equation y = 2x² - 3 represents a parabola.
• The parabola opens upwards.
• The vertex of the parabola is at the point (0, -3).
• The parabola is narrower than y = x².

Now that we’ve thoroughly examined both equations individually, let’s bring them together and discuss how they interact within the system.

Describing the System as a Whole

Alright, guys, we've dissected the circle and the parabola separately. Now, let's put the pieces together and describe the system of equations as a whole. This means understanding how these two curves interact when they're graphed on the same coordinate plane.

Visualizing the Intersection

Imagine plotting both the circle and the parabola on the same graph. We have a circle centered at (0, 0) with a radius of √2, and a parabola opening upwards with its vertex at (0, -3). The key question is: how do these two shapes intersect?

To visualize this, think about the circle sitting above the x-axis, and the parabola’s vertex sitting below the x-axis at (0, -3). As the parabola opens upwards, it will eventually intersect the circle. The number of intersection points tells us how many solutions the system of equations has.

Predicting the Number of Solutions

Based on the shapes and positions of the circle and parabola, we can anticipate that they will intersect at two points. Why? The parabola opens upwards from (0, -3), and the circle is centered at (0, 0) with a radius of approximately 1.414. This geometry suggests two clear intersection points – one on the left side of the y-axis and one on the right.

Graphical Representation

A graphical representation is super helpful here. If you were to plot these equations using graphing software or even by hand, you’d see the two points of intersection clearly. This visual confirmation solidifies our understanding of the system's behavior.

Solving the System (Briefly)

While this article focuses on describing the system, it's worth mentioning that you could solve this system algebraically to find the exact coordinates of the intersection points. This involves substituting one equation into the other and solving the resulting equation. However, for our descriptive purposes, understanding the geometric interaction is key.

Key Descriptors of the System

• The system consists of a circle centered at (0, 0) with a radius of √2 and a parabola opening upwards with a vertex at (0, -3).
• The circle and parabola are expected to intersect at two points.
• The system represents a non-linear system of equations, as it involves equations of degree two.

Why This Matters

Describing a system of equations isn't just about identifying shapes; it's about understanding the relationships between equations. It’s a foundational skill in mathematics that extends to many fields, including physics, engineering, and computer science. Being able to visualize and describe these systems allows us to predict their behavior and solve real-world problems.

Final Thoughts

So, guys, when you encounter a system of equations, take a moment to describe what you’re seeing. Identify the shapes, their key features, and how they interact. This descriptive approach is a powerful tool for understanding and solving mathematical problems.

Conclusion

In this article, we've thoroughly explored how to describe a system of equations, using the specific example of x² + y² = 2 and y = 2x² - 3. We identified the first equation as a circle centered at the origin with a radius of √2, and the second as a parabola opening upwards with a vertex at (0, -3). By understanding the characteristics of each equation and their graphical representation, we were able to predict that the system would have two solutions.

Describing systems of equations is a fundamental skill in mathematics. It allows us to visualize and understand the relationships between different equations, which is crucial for solving problems in various fields. Remember, guys, breaking down complex problems into smaller, understandable parts is always the key to success.

Keep practicing, keep exploring, and you'll become a system-describing pro in no time! Thanks for joining me on this mathematical journey!