The image shows a student writing a rule to describe a transformation. The rule is: "Reflect across the y-axis, then shift up 3 units." This rule would take the point M to point J.
Reflection across the x-axis:
Imagine the x-axis as a mirror. Every point in the graph is reflected across this mirror to its opposite side.
For example, the point (3,2) would be reflected to (3,−2), and the point (−1,4) would be reflected to (−1,−4).
This rule works for any point above or below the x-axis, but it wouldn't work for points that lie directly on the x-axis itself.
2. Vertical stretch by a factor of 2:
Think of inflating a balloon vertically. Every point in the graph is stretched vertically by a factor of 2, making it twice as far from the x-axis.
For example, the point (2,1) would be stretched to (2,2), and the point (4,−3) would be stretched to (4,−6).
This rule works for all points in the graph, regardless of their position relative to the x-axis or the y-axis.
3. Translation down 1 unit:
Imagine the entire graph being shifted down one unit. Every point moves down 1 unit in the y direction, without changing its position in the x direction.
For example, the point (3,2) would be translated to (3,1), and the point (−1,4) would be translated to (−1,3).
This rule works for all points in the graph, regardless of their position relative to the x-axis or the y-axis.
Combined transformation:
The most likely rule that describes the transformation in the image is a combination of these three individual transformations:
Reflection across the x-axis
Vertical stretch by a factor of 2
Translation down 1 unit
This combination of transformations would map each point in the original graph to its corresponding point in the transformed graph.