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Function shifts are transformations that change the position of a graph on a plane without altering its shape. There are two types of function shifts: horizontal and vertical.

• A vertical shift involves moving the graph up or down. If you add a constant to the function, such as in the equation \( f(x) + k \), the graph shifts upward by \( k \) units. Conversely, subtracting a constant, as in \( f(x) - k \), shifts the graph downward by \( k \) units.
• A horizontal shift involves moving the graph left or right. This is done by modifying the input variable \( x \) in the function. For example, replacing \( x \) with \( x - h \) in \( f(x) \) will shift the graph to the right by \( h \) units. On the other hand, using \( x + h \) shifts the graph left by \( h \) units.

This can be an essential tool in adjusting how the graph aligns with your data or fits the context of a problem you are solving. Every shift affects the position of the graph while maintaining the original curve or line's shape.

Reflections are another kind of transformation that flips a graph over a specified axis. This transformation can alter the appearance of the graph significantly while preserving its overall structure.

• A reflection about the \( x \)-axis involves taking each point of a function \( f(x) \) and reflecting it over the \( x \)-axis. This is achieved by multiplying the entire function by \(-1\), resulting in \(-f(x)\). As a result, this transformation changes the sign of all the \(y\)-values of the graph, effectively flipping it upside down.
• A reflection about the \( y \)-axis occurs when the graph is mirrored over the vertical \( y \)-axis. This is accomplished by replacing \( x \) with \(-x\) in the function, yielding \( f(-x) \). This reflection changes the direction of the graph horizontally, so that the left side and the right side are swapped.

Reflections are useful for creating symmetry in graphs or for analyzing functions' behaviors when flipped.

A vertical stretch is a transformation that elongates a graph in the vertical direction. This change affects the graph's height but retains its horizontal attributes.
To stretch a graph vertically, you multiply the function by a factor greater than one. For instance, if your function is \( f(x) \) and you multiply it by 3, you will get \( 3f(x) \). This means that every \( y \)-value in the graph will be three times its original size, effectively stretching the graph upward.

• For example, if a point was originally at \((1, 2)\), after a vertical stretch by a factor of 3, the point's position would move to \((1, 6)\).

An important thing to note is that vertical stretches do not change the \(x\)-values of the graph, so the width remains constant. This transformation is particularly useful when you want to emphasize the vertical aspect of a graph without changing its horizontal positioning.

Vertical shrink is another transformation that modifies a graph but compresses it vertically instead of stretching it.

• To shrink a graph vertically, you multiply the function by a factor between 0 and 1. For instance, multiplying the function \( f(x) \) by \( \frac{1}{3} \) results in \( \frac{1}{3}f(x) \), reducing each \( y \)-value to one-third of its original value.

This transformation effectively pulls the graph closer to the x-axis.
Unlike horizontal changes, in vertical shrink, the \(x\)-values remain unchanged, keeping the graph's width the same.
Overall, vertical shrinking can help adjust the graph to fit a particular scale or to compare multiple graphs on the same plot where varying scales are present.