Describe how each graph is obtained from the graph of 𝔂 = ƒ(x).


e. 𝔂 = ƒ( x ) - 4
3

Verified step by step guidance

1. Start with the graph of the function 𝔂 = ƒ(x). This is your base graph from which transformations will be applied.
2. The expression 𝔂 = ƒ(x) - 4 indicates a vertical shift. Specifically, subtracting 4 from the function means you will shift the entire graph of 𝔂 = ƒ(x) downward by 4 units.
3. To visualize this, take each point (x, y) on the original graph of 𝔂 = ƒ(x) and move it to the point (x, y - 4). This will lower every point on the graph by 4 units.
4. Ensure that the shape and orientation of the graph remain unchanged; only the vertical position is altered.
5. After applying the vertical shift, the new graph represents the function 𝔂 = ƒ(x) - 4, which is the original graph moved down by 4 units.

Key Concepts

Vertical Shifts

Vertical shifts occur when a constant is added to or subtracted from a function. In the case of 𝔶 = ƒ(x) - 4, the graph of ƒ(x) is shifted downward by 4 units. This transformation affects the y-coordinates of all points on the graph, while the x-coordinates remain unchanged.

Function Notation

Function notation, such as ƒ(x), represents a relationship where each input x corresponds to exactly one output y. Understanding this notation is crucial for interpreting how changes to the function, like subtracting a constant, affect the overall graph. It allows for clear communication of mathematical ideas and transformations.

Graph Transformations

Graph transformations refer to the various ways a function's graph can be altered, including shifts, stretches, and reflections. In this case, the transformation involves a vertical shift, which is a fundamental concept in understanding how the graph of a function can be manipulated without changing its shape.