![]() It is for students from Year 7 who are preparing for GCSE. This is a KS3 lesson on reflecting a shape in the line y x using Cartesian coordinates. Looking again at A$^\prime$ and A$^$ its midpoint lies at the origin (0,0), and the same is true for all other points. This page includes a lesson covering how to reflect a shape in the line y x using Cartesian coordinates as well as a 15-question worksheet, which is printable, editable and sendable. The same calculations work for the other points: in each case, the $x$-coordinate does not change and the $y$-coordinate changes sign.īelow is a picture of the original points, their reflections over the $x$-axis and then the reflections of the new points over the $y$-axis: ![]() If we were to fold the plane along the $x$-axis, the points A and A$^\prime$ match up with one another. Reflecting over the $x$-axis does not change the $x$-coordinate but changes the sign of the $y$-coordinate. Similarly the coordinates of $B$ are $(-4,-4)$ while $C = (4,-2)$ and $D = (2,1)$.īelow is a picture of the reflection of each of the four points over the $x$-axis: The coordinates of $A$ are $(-5,3)$ since $A$ is five units to the left of intersection of the axes and 3 units up. In order to help identify patterns in how the coordinates of the points change, the teacher may suggest for students to make a table of the points and their images after reflecting first over the $x$-axis and then over the $y$-axis: Point Thus the knowledge gained in this task will help students when they study transformations in the 8th grade and high school. Later students will learn that this combination of reflections represents a 180 degree rotation about the origin. This means that if we reflect over the $x$-axis and then the $y$-axis then both coordinates will change signs. Similarly when we reflect a point $(p,q)$ over the $y$-axis the $y$-coordinate stays the same but the $x$-coordinate changes signs so the image is $(-p,q)$. ![]() When we reflect a point $(p,q)$ over the $x$-axis, the $x$-coordinate remains the same and the $y$- coordinate changes signs so the image is $(p,-q)$.The teacher may wish to prompt students to identify patterns in parts (b) and (c): Then consider how you think reflections would work in $\Bbb R^n$ for other values of $n$.The goal of this task is to give students practice plotting points and their reflections. Try to visualize each of these reflections in $\Bbb R^2$ and $\Bbb R^3$. Go back and look up the geometric properties of even and odd functions if you don't remember how these reflections work in $\Bbb R^2$ (note however that you can still reflect through the origin in $\Bbb R^3$). The difference between reflecting through a line vs a plane in $\Bbb R^3$ is comparable to reflecting through the origin vs a line in $\Bbb R^2$. Let's see how this affects the standard basis $\$$ Corresponding parts of the figures are the same distance from the line of reflection. The far end of that line segment is then at the point that is the reflection of your point across the $y$-axis. To perform a geometry reflection, a line of reflection is needed the resulting orientation of the two figures are opposite. Extend that line segment past $y$ by the same length as the distance from the point to the $y$-axis. Now connect that point to the $y$-axis by a line segment that is orthogonal to the $y$-axis. Write the notation to describe this reflection for Thomas. Thomas describes a reflection as point Jmovingfrom\ (J( 2, 6) to J ( 2, 6). Consider an arbitrary point in $\Bbb R^3$. To write a rule for this reflection you would write: rx axis(x, y) (x, y). ![]() ![]() Unfortunately I can't find a good image on Google Images to describe reflection through a line in $\Bbb R^3$ (and my pgfplots-fu is still pretty basic), but I'll try to describe what it means. There are three basic transformations that can be applied to graphs of linear functions: sliding the line around (translation), flipping the line (reflection), and stretching the line (scaling. When a question asks you to find a matrix representing a linear transformation $T$ that is only described geometrically, your task is to figure out how that $T$ transforms a basis for your domain. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |