Question

in the coordinate plane continued
6 quadrilateral pqrs is transformed in the coordinate plane to form the image pqrs. label the vertices of pqrs and tell what single transformation was used.
7 triangle fgh is transformed in the coordinate plane to form the image fgh. label the vertices of fgh and tell what single transformation was used.
draw a figure and an image of that figure in such a way that any of the three transformations (a single translation, a single reflection, or a single rotation) could be used to obtain it.

Explanation:

Step1: Analyze the transformation of PQRS

By observing the coordinates of PQRS and P'Q'R'S', we can see that the x - coordinates of the vertices change in a certain pattern and the y - coordinates change in a certain pattern. Comparing the original and new coordinates, we find that the transformation is a reflection.

Step2: Determine the type of reflection

For a reflection over the x - axis, the rule is (x,y)→(x, - y), for y - axis (x,y)→(-x,y). By looking at the change in signs of coordinates of PQRS to P'Q'R'S', we find that it is a reflection over a line. In this case, it is a reflection over the line y = - x. The rule for reflection over y=-x is (x,y)→(-y,-x). For example, if P has coordinates (x1,y1) and P' has coordinates (y1,x1) with appropriate signs changed according to the rule.

Step3: Analyze triangle FGH

We need to find the vertices of F'G'H' and the transformation. First, we would need to know the original coordinates of F,G,H. Then, by comparing with the new - formed triangle's vertices (which are not given numerically here but can be found by observing the grid), we can determine the transformation. If we assume the original coordinates of F(xF,yF), G(xG,yG), H(xH,yH) and find the new coordinates F'(xF',yF'), G'(xG',yG'), H'(xH',yH') by looking at the grid. If the x and y coordinates change signs and swap positions, it is a reflection over y = - x. If only the x - coordinates change signs, it is a reflection over the y - axis. If only the y - coordinates change signs, it is a reflection over the x - axis.

Step4: Draw a figure and its image

For drawing a figure and its image, we can draw a square ABCD with vertices A(1,1), B(1,2), C(2,2), D(2,1). For a translation, we can move it 3 units to the right and 2 units up, so A' would be (4,3), B' would be (4,4), C' would be (5,4), D' would be (5,3). For a reflection over the x - axis, A'' would be (1, - 1), B'' would be (1, - 2), C'' would be (2, - 2), D'' would be (2, - 1). For a 180 - degree rotation about the origin, the rule is (x,y)→(-x,-y), so A''' would be (-1,-1), B''' would be (-1,-2), C''' would be (-2,-2), D''' would be (-2,-1).

Answer:

For quadrilateral PQRS, the vertices of P'Q'R'S' are as labeled (P'(4, - 1), Q'(8, - 2), R'(6, - 5), S'(2, - 5)) and the transformation is a reflection over the line y = - x. For triangle FGH, find the vertices of F'G'H' by observing the grid and comparing with the original triangle's vertices to determine the transformation (it could be a reflection over x - axis, y - axis or y=-x depending on the coordinate changes). For drawing a figure and its image, an example is given above with a square and its translated, reflected and rotated images.