Introduction
Imagine that you are looking at an elevation map. This map’s purpose is to show the various elevations in a specific region. The different elevations are distinctly marked by lines that are called contour lines. These contour curves for the function gives the elevation of the land in that area. This is the general idea of level curves. Level curves are a cross section of a graph at a specific value. Imagine going in person to a specific region that you looked at on an elevation map. You would easily be able to see how the elevation changes at each contour line. Now, if you were to theoretically take layers of this region (like a cake) you would then be able to visualize the level curves of the region. Today, we are going to dive further into level curves by analyzing a function and finding eight distinct level curves on the graph of the function.
Level Curves?
To further understand the concept of level curves, we need to first understand that we are going to be working with a function in the three-dimensional space. Since we will be working in the third dimension, level curves will have the function \(z=f(x,y)\). This will produce a two-dimensional curve that we would get by setting \(z=k\), where \(k\) is some number. This will give us the equation of the level curves to be \(f(x,y)=k\). Without looking at an example yet, lets understand the process on how to visualize level curves. As we find the different level curves at different values of \(k\), we can graph them in the second dimension to visualize how the function behaves. This will allow us to have a general visualization of how the solid will look in the third dimension. Now that we have a brief understanding of how level curves work, we can get into the example.
Visualizing Level Curves
Today, we are going to be looking at the function: \[f(x,y)=y^2-2x^2\] To visualize how this function really acts, we are going to be looking at the values of \(k\) when \(k\) is both positive and negative. To do this, we are going to set the function equal to \(k\) which gives the function: \(y^2-2x^2=k\). From this, we can find different values of \(k\) and find the respective level curve at that value. Below is a Desmos graph that shows how the function acts when \(k\) is both positive and negative. I chose the \(k\) values when \(k = -4, -3, -2, -1, 0, 1, 2, 3, 4 \)
Now from the graph, since we are only in the second dimension, we cannot exactly see how the different values of \(k\) actually impacted the region. Before we visualize the solid in the third dimension, lets understand how the different values of \(k\) impact the region. When the \(k\) value is negative, that means that the solid is going in the negative z direction. When the \(k\) value is positive, that means that the solid is going in the positive z direction. If we revisit the idea of an elevation map, we can visualize this a bit better. When we travel further out from the blue lines we will increase in elevation, and when we travel further out from the red lines we will decrease in elevation. Now that we have a further understanding of the object, we can visualize it in the third dimension. 
This image is only a 3D visualization of our function and doesn't include our actual level curves. From the previous graph, we found the level curves of the function and now we basically want to add the two together. Below is the final image that includes the third dimensional model with the included level curves of the solid. 
Why These Functions?
There are a few different reasons on why I chose the functions that I did. First, I thought that the function did a great job at incorporating the positive and negative direction of the z-axis. This would allow for someone new to the topic to grasp the concept a bit better. Also, I really like the shape that the function produced. It was one of the best functions that I was experimenting with that represented a saddle the most. Lastly, this function allowed for someone to easily visualize the concept of level curves and how they work in both the positive and negative direction.
Imagine that you are looking at an elevation map. This map’s purpose is to show the various elevations in a specific region. The different elevations are distinctly marked by lines that are called contour lines. These contour curves for the function gives the elevation of the land in that area. This is the general idea of level curves. Level curves are a cross section of a graph at a specific value. Imagine going in person to a specific region that you looked at on an elevation map. You would easily be able to see how the elevation changes at each contour line. Now, if you were to theoretically take layers of this region (like a cake) you would then be able to visualize the level curves of the region. Today, we are going to dive further into level curves by analyzing a function and finding eight distinct level curves on the graph of the function.
Level Curves?
To further understand the concept of level curves, we need to first understand that we are going to be working with a function in the three-dimensional space. Since we will be working in the third dimension, level curves will have the function \(z=f(x,y)\). This will produce a two-dimensional curve that we would get by setting \(z=k\), where \(k\) is some number. This will give us the equation of the level curves to be \(f(x,y)=k\). Without looking at an example yet, lets understand the process on how to visualize level curves. As we find the different level curves at different values of \(k\), we can graph them in the second dimension to visualize how the function behaves. This will allow us to have a general visualization of how the solid will look in the third dimension. Now that we have a brief understanding of how level curves work, we can get into the example.
Visualizing Level Curves
Today, we are going to be looking at the function: \[f(x,y)=y^2-2x^2\] To visualize how this function really acts, we are going to be looking at the values of \(k\) when \(k\) is both positive and negative. To do this, we are going to set the function equal to \(k\) which gives the function: \(y^2-2x^2=k\). From this, we can find different values of \(k\) and find the respective level curve at that value. Below is a Desmos graph that shows how the function acts when \(k\) is both positive and negative. I chose the \(k\) values when \(k = -4, -3, -2, -1, 0, 1, 2, 3, 4 \)
There are a few different reasons on why I chose the functions that I did. First, I thought that the function did a great job at incorporating the positive and negative direction of the z-axis. This would allow for someone new to the topic to grasp the concept a bit better. Also, I really like the shape that the function produced. It was one of the best functions that I was experimenting with that represented a saddle the most. Lastly, this function allowed for someone to easily visualize the concept of level curves and how they work in both the positive and negative direction.
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