![]() The image of $g$ is $\triangle A^$ on the other, these triangles are not similar and so their angles are not congruent. The image of $f$ is $\triangle A^\prime B^\prime C^\prime$ which is congruent to the original triangle but it is displaced to the left by 3 units. ![]() The teacher should prompt students for a geometric description of what $f$ and $g$ do so that students understand what transformations of the plane these functions represent.īelow is a picture of the images of the triangle after applying the maps $f$ and $g$: Graph ABC with vertices A(3, 2), B(6, 3), and C(7, 1) and its image after the glide reflection. ![]() The task does not specify that $f$ is a translation to the left by 3 units and that $g$ is a horizontal stretch by a factor of 3. The rule is: (x, y) (x + 3, y 3) Mikah describes a translation as point D in a diagram moving from D(1, 5) to D(3, 1). In the solution a similarity argument is provided but students familiar with trigonometry could also calculate the angles in the different triangles and verify in this way that horizontal stretch changes the angles of this particular triangle. The fact that horizontal stretch does not preserve angles is visibly clear but requires more careful thinking. This means, we switch x and y and make x negative. The only line segments whose distance is preserved by this horizontal stretch are vertical lines and the teacher may wish to have students investigate this as a follow up question. In today’s geometry lesson, we’re going to review Rotation Rules. The fact that horizontal stretch does not preserve distances can be seen from the pictures or using the Pythagorean theorem. Earlier, you were told that Jack described a translation as point J moving from J(2, 6) to J'(4, 9). Positive y translates upwards, negative y translates downwards.The goal of this task is to compare a transformation of the plane (translation) which preserves distances and angles to a transformation of the plane (horizontal stretch) which does not preserve either distances or angles. So: D(1, 4) D'(1 + a, 4 + b) or D(1, 4) D'(6, 1) Therefore: 1 + a 6 a 7 and 4 + b 1 b 3. It's at x equals negative seven, if you move eight to the right, if you increase your x coordinate by eight, you're gonna move to x equals one, and then if you change your y coordinate by negative one, you're gonna move down one, then you're gonna get to that point right over there.Positive x translates to the right, negative x translates to the left.Always remember the translation is the final position minus the start position, and double check that the signs are consistent with the rules: If we compare the top points of the two triangles, we can see that the translation distance is 5.Ī second common mistake is to get the signs of the translation vector incorrect. This distance is 2.īut that distance isn't the translation distance, because we are not using the equivalent points on each shape. In this diagram, we have marked the distance from the rightmost point of A to the leftmost point of B. ![]() Show the result of translating this shape:Ī common mistake is to use the gap between the shapes rather than the distance the shape has been translated: The (x, y) coordinates of these points are (x, y) (x + 1, y + 4) (2, 0), (5, 7). For D(2, 3), the translated coordinate will be (x-0, y-5) (2-0, 3-5) Hence, (2, -2) is a translated coordinate. For C(-5, 3), the translated coordinate will be (x-0, y-5) (-5-0, 3-5) Hence, (-5, -2) is a translated coordinate. The shape is moved 4 units to the left and 5 units up, so the translation vector is:ĭescribe the single transformation that maps shape A onto shape B: For B(2, 7), the translated coordinate will be (x-0, y-5) (2-0, 7-5) Hence, (2, 2) is a translated coordinate. The shape is moved 3 units to the right and 4 units up, so the translation vector is: In geometry, a reflection is a rigid transformation in which an object is mirrored across a line or plane. This example shows a rectangle translated in the x and y directions: Here are some simple things we can do to move or scale it on the graph: We can move it up or down by adding a constant to the y-value: g(x) x 2 + C. Rule: A positive y translation moves the shape upwards, and a negative y translation moves the shape downwards. Just like Transformations in Geometry, we can move and resize the graphs of functions: Let us start with a function, in this case it is f(x) x 2, but it could be anything: f(x) x 2.
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