### Coordinate Transformations #### Understanding the coordinates of points after successive transformations: 1. **The coordinates of N' after the first transformation:** - **?** = (-1, 3) 2. **The coordinates of N'' after the second transformation:** - **?** = (5, -1) 3. **The coordinates of M' after the first transformation:** - **?** = (3, 1) 4. **The coordinates of M'' after the second transformation:** - **?** = (3, -2) 5. **The coordinates of R' after the first transformation:** - **?** = (2, -3) 6. **The coordinates of R'' after the second transformation:** - **?** = (1, 5) #### Graphs and Diagrams: This section presents information in a tabular format addressing the coordinates of specific points (N, M, R) after one or more transformations. Each original point is subjected to a series of transformations resulting in new coordinate points, labeled with prime (') and double prime ('') notations to denote each successive transformation stage. Ensure to visualize or plot these coordinates in a Cartesian plane for better understanding and graphical representation of transformations. ### Example 7: Complete the Following Compositions **Task:** Apply the following transformations to a triangle: 1. **Translation** \((x, y) \rightarrow (x + 7, y + 2)\) 2. **Counterclockwise Rotation of 90°** **Given:** The preimage is a triangle with vertices at points \(M(-2, -3)\), \(N(-4, -1)\), and \(R(-5, -5)\). **Hint:** The rotation rules are in the 7.3 notes. --- **Graph Explanation:** - The graph provided shows a Cartesian coordinate system. - The triangle preimage is plotted: - Point \(M\) is located at \((-2, -3)\). - Point \(N\) is located at \((-4, -1)\). - Point \(R\) is located at \((-5, -5)\). The vertices \(M\), \(N\), and \(R\) form a triangle, and the preimage is clearly labeled on the graph. To proceed with the task: 1. Perform the translation \((x + 7, y + 2)\) on each vertex. 2. Apply a counterclockwise rotation of 90° to the translated coordinates. Please refer to your 7.3 notes for rotation rules to complete the task accurately.
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
I tried my best to get it in one picture but I really need help it say to match the coordinates with the translation
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