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When we talk about graph reflections, we're examining how the shape of a graph changes by flipping it across an axis. Imagine holding a mirror along either the x-axis or the y-axis: if we reflect over the y-axis, every point on the graph moves horizontally across the axis to a position that is equidistant from the axis but on the opposite side. For instance, a point \( (a, b) \) when reflected over the y-axis becomes \( (-a, b) \). This is what happens in the transformation \( y = f(-x) \). It's as if you take your graph and flip it left to right, maintaining the original height of each point on the graph. Similarly, reflecting over the x-axis takes a point and flips it vertically, changing \( (a, b) \) into \( (a, -b) \). In this case, you are turning your graph upside down. This double reflection is seen in transformations like \( y = -f(-x) \). Understanding these reflections is crucial for grasping how a function's graph can change form without changing its size or shape.

Vertical shifts involve moving every point on a function's graph up or down by a certain number of units. Consider it like sliding the entire graph up or down along the y-axis. If we add a positive value to the function, like in \( y = f(x) + 2 \), we shift the graph upward by 2 units. Every point on the graph finds itself 2 units higher while maintaining its horizontal position. Conversely, subtracting a value will shift the graph downward by that same amount. For example, in the equation \( y = -f(x) + 1 \), each point is moved 1 unit up after the vertical reflection. This reflects how adding to or subtracting from the function generally affects how it translates along the vertical axis.

Horizontal shifts move the entire graph to the left or right past its original position. This type of transformation involves changing the x-coordinates of all the points, while the y-coordinates stay the same. For example, in the case of \( y = f(x - 3) \), every point on the graph shifts 3 units to the right because of the negative sign before 3, giving \( (a+3, b) \). Alternatively, \( y = f(x + 1) \) shifts the graph 1 unit to the left, yielding a transformed point of \( (a-1, b) \). It's essential to note that adding to x inside the function shifts the graph left, while subtracting shifts it right. This might seem opposite to intuition but is a key characteristic of horizontal shifts.

Point transformations combine different movements of points on the graph, involving reflections, and vertical and horizontal shifts all at once. Each of these adjustments affects the point's final position. For example, in the point transformation corresponding to \( y = -f(1-x) + 1 \), multiple steps are applied:

• First, reflect the graph over the y-axis, moving \( (a, b) \) to \( (1-a, b) \).
• Then, flip it over the x-axis, resulting in \( (1-a, -b) \).
• Finally, add 1 to achieve a vertical shift upwards, turning the point into \( (1-a, -b+1) \).

By combining these transformations, the graph is altered in a complex way, involving both shifts and flips, which collectively create the final graph.