In this section, we will introduce some examples of graphs in two dimensions that have nice algebraic properties and a variety of interesting geometric features. For our preview activity, we will recall some properties of graphs and equations that will be useful in describing our new examples. While students may have different levels of experience with graphs like circles, parabolas, ellipses, and hyperbolas, we will focus our work on the correspondence between the algebraic presentations of these shapes and the important geometric features.
We have seen at the end of Section 9.1 how the distance formula leads to the equation of a sphere and how the idea of flat one and two dimensional graphs gave rise to lines (Section 9.5) and planes (Section 9.6). Here we introduce some other curves that are used throughout the rest of this text as examples. These objects will allow us to have graphs with a variety of geometric features while still being algebraically simple. You may already be familiar with the curves studied in this section, in which case you can consider this section a review. However, if you have not studied these curves before, we have designed this section to help you understand their key properties.
In this activity we will look at how to translate coordinate systems, which means to use a new coordinate system with axes parallel to to the original set of axes and the same scale. In other words, the new coordinate system has a new origin but does not change the way coordinates are measured. This is useful to simplify the equation of a graph by making the graph’s center the new origin.
On the axes below, draw and label the point \(P=(4,-1)\text{.}\) With a different color, draw a new set of axes that are centered at \(P\) and label these axes \(x^*\) and \(y^*\text{.}\) The \(x^*\)- and \(y^*\)-axes should parallel to the \(x\)- and \(y\)-axes and use the same scale.
Generalize your work for part b and write \(x^*\) and \(y^*\) in terms of \(x\text{,}\)\(y\text{,}\) and the coordinates of new center \((4,-1)\text{.}\)
While walking your dog, you notice that your dog takes three steps for every step of yours. When you sit down at home you realize that your coordinate system for your house is different than your dog’s. You decide to make the origin of both your coordinate system and your dog’s to be the entrance to your kitchen. Your dog’s bed is eight of your steps south and four steps west of the entrance to the kitchen. The dog’s water bowl is 5 of your steps north and 3 steps east from the entrace to your kitchen. The front door of your house is 15 of your steps south and 9 steps east of the entrance to the kitchen.
Draw a set of axes and plot the location of your dog’s bed, water bowl, and the front door in terms of your steps from the origin (the entrance to your kitchen).
If your dog’s favorite toy is 17 dog steps east and 5 dog steps north of the entrance to the kitchen and your dog’s collar is 13 dog steps south and 7 dog steps west, give the coordinates of your dog’s toy and collar in terms of your steps.
If you call the coordinates in your steps the \((x,y)\)-coordinate system and refer to coordinates in your dog’s steps as the \((x',y')\)-coordinate system, give the location in both coordinate systems for your dog’s bed, water bowl, the front door, your dog’s favorite toy, and your dog’s collar.
The results of the previous activities are probably not suprising but should offer some insight into why some aspects of coordinate transformations seem backwards. This is because the transformations are about converting back to the measurement of a different coordinate system.
You may have seen the ideas of coordinate transformation that were central to Activity A.1.1, but hopefully you see precisely where the algebraic transformations come from and how they correspond to simple geometric transformations. For our basic shapes in two dimensions, we will give a brief definition for the shape, but the focus of this presentation will be about applying transformations to a basic shape in order to generalize the possible usage of these shapes.
A circle is the set of points that are a fixed distance (called the radius) away from a specific point (called the center). Most often, circles are introduced with the center at the origin and the radius given by a constant \(R\text{.}\) This means that a point \((x,y)\) that is on the circle will satisfy
which is often called the standard form of the circle. The standard form is convenient to use because the information needed to graph the circle can be read from this form without needing more algebra. For instance, the circle given by
has center \((2,-3)\) and radius \(\sqrt{6}\text{.}\) Notice that the transformation equations from Activity A.1.1 show up in this example as we move a circle from being centered at the origin, where the equation is \(x^2+y^2=R^2\text{,}\) to the point \((h,k)\text{,}\) where the equation is \((x-h)^2+(y-k)^2=R^2\text{.}\)
An ellipse is the set of all points such that the sum of the distances from the point \((x,y)\) to a pair of distinct points (called foci) is a fixed constant.
To help illustrate the definition of an ellipse, explore Figure A.1.2. You can use the slider to change the location on the ellipse being displayed. You should see how as you move around the ellipse, the sum of the lengths of the blue and red segments remains constant, as shown compared to the green segment, which is a fixed length.
While the definition of the ellipse given above has applications in engineering, orbital mechanics, and optics, we will focus on the ellipse as a transformation of a circle. While this is not obvious and the details take a little while to prove, any ellipse in the plane can be obtained by transforming a circle (through translation, horizontal/vertical coordinate stretches that are not the same as each other, and rotations of the coordinate systems).
What transformation is done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{x^2}{4}+y^2=1\text{?}\) You should be specific about how the graph of \(\frac{x^2}{4}+y^2=1\) is different than the graph of \(x^2+y^2=1\text{.}\)
What transformations are done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{x^2}{4}+\frac{y^2}{9}=1\text{?}\) You should be specific about how the graph of \(\frac{x^2}{4}+\frac{y^2}{9}=1\) is different than the graph of \(x^2+y^2=1\) and specify if the transformations need to be done in a particular order.
What transformations are done to convert between the circle given by \(x^2+y^2=1\) and the graph of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\text{?}\) You should be specific about how the graph of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\) is different than the graph of \(x^2+y^2=1\) and specify if the transformations need to be done in a particular order.
Draw a plot of \(\frac{(x+2)^2}{4}+\frac{(y-3)^2}{9}=1\) and label the center of your plot and the points that demonstrate how far the ellipse is stretched in the vertical and horizontal directions.
The graph of the equation \(9x^2+16y^2=400\) is an ellipse. Convert this equation to the form \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this ellipse and label all extreme points on your plot.
The graph of the equation \(4x^2+y^2+24x-2y+21=0\) is an ellipse. Convert this equation to the form \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this ellipse and label all extreme points on your plot.
Having considered the curves where the sum of distances to two points is constant, we now turn our attention to curves where the difference of the distances is constant.
A hyperbola is the set of all points such that the difference of the distances from the point \((x,y)\) to a pair of distinct points (called foci) is a fixed constant.
In Figure A.1.5, you can see how as you move around the hyperbola, the difference of the lengths of the blue and red segments remains constant, as shown in green. Additionally, you can see that a hyperbola has asymptotic behavior in that as you approach the edges of the plot, the hyperbola will get very close to the dashed asymptote lines.
The most basic equation for a hyperbola is \(x^2-y^2=1\text{.}\) Make a plot of the hyperbola given by \(x^2-y^2=1\text{.}\) In your plot, be sure to include the asymptotes, which are given by \(y=\pm x\text{,}\) as well as the vertices. These are the points on the hyperbola that are closest to the center. In this case, the vertices are the \(x\)-intercepts.
What transformation is done to change the graph of the hyperbola given by \(x^2-y^2=1\) to the graph of \(\frac{x^2}{4}-y^2=1\text{?}\) Be specific about how the graph of \(\frac{x^2}{4}-y^2=1\) is different than the graph of \(x^2-y^2=1\text{.}\)
What transformation is done to change the graph of the hyperbola given by \(x^2-y^2=1\) to the graph of \(y^2-x^2=1\text{?}\) Be specific about how the graph of \(y^2-x^2=1\) is different than the graph of \(x^2-y^2=1\text{.}\)
What transformations are done to change the graph of the hyperbola \(x^2-y^2=1\) to the graph of \(\frac{x^2}{4}-\frac{y^2}{9}=1\text{?}\) Be specific about how the graph of \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is different than the graph of \(x^2-y^2=1\) and specify if the transformations need to be done in a particular order.
Draw a graph of \(\frac{(x+2)^2}{4}-\frac{(y-3)^2}{9}=1\) and label the center, the vertices, and the asymptote lines for the hyperbola. You will need to apply the transformations from the previous part to the asymptotes \(y = \pm x\) of the base hyperbola in order to get the equations of the transformed asymptotes.
The graph of the equation \(9x^2-16y^2=400\) is a hyperbola. Convert this equation to the form \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this hyperbola and label the center, the vertices, and the asymptote lines of the hyperbola.
The graph of the equation \(4x^2-y^2+24x-2y+21=0\) is an hyperbola. Convert this equation to the form \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\text{.}\) Then use the idea of transformations from above to graph this hyperbola and label the center, the vertices, and the asymptote lines of the hyperbola.
In Figure A.1.8, you can see how as you move around the parabola, the lengths of the blue and red segments remain the same as each other. The point that is halfway between the focus and the directrix is referred to as the vertex of the parabola.
Draw a plot of the graph for \(2x=y^2\) and label the vertex and four other points. What transformation is done to change the graph of the parabola given by \(x=y^2\) to the graph of \(2x=y^2\text{?}\) Be specific about how the graph of \(2x=y^2\) is different than the graph of \(x=y^2\text{.}\)
Draw a plot of the graph for \(y=x^2\) and label the vertex and four other points on the parabola. What transformation is done to change the graph of the parabola given by \(y=x^2\) to the graph of \(x=y^2\text{?}\) Be specific about how the graph of \(x=y^2\) is different than the graph of \(y=x^2\text{.}\)
What transformations are done to change the graph of \(x=y^2\) to the graph of \(\frac{x-1}{2}=\left(\frac{y+2}{3}\right)^2\text{?}\) Be specific about how the graph of \(\frac{x-1}{2}=\left(\frac{y+2}{3}\right)^2\) is different than the graph of \(x=y^2\) and specify if the transformations need to be done in a particular order.
The graph of the equation \(x^2-8x-8y+8=0\) is an parabola. Convert this equation to the form \(\left(\frac{x-h}{a}\right)^2=\frac{y-k}{b}\) and use the idea of transformations from above to graph this parabola. Label the vertex and four other points on the parabola.
Hyperbola: \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\) with center \((h,k)\text{,}\) vertices \((h\pm a,k)\text{,}\) and asymptotes \((y-k)=\frac{b}{a}(x-h)\)
SubsectionA.1.4Notes to Instructors and Dependencies
This section includes basic transformation ideas that students may have a range of experience with. We included these activities as a way to make sure that students have some exposure and they get to practice simple algebraic procedures that will be used with more complexity later in the text. A more thorough approach to the conic sections can be found at (insert link to IBL materials here). You may want to have students work through activities like Activity A.1.1 and Activity A.1.2 on their own during times where you don’t have a preview activity to prepare for new material.
