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In mathematics, a horizontal shift is a function transformation that moves the graph of a function left or right. This type of shift does not alter the shape of the graph; it only relocates it along the horizontal axis. To determine a horizontal shift, focus on the change inside the function's argument.

Consider the function shown as \[ y = f(x - c) \].

Here, the parameter \( c \) indicates the number of units the graph will be shifted to the right if \( c \) is positive, and to the left if \( c \) is negative. For our exercise, given the expression \( f(x-4) \), we see a horizontal shift of the graph of function \( f \) to the right by 4 units. This means every point on the graph will move 4 units to the right. Specifically, starting with a point \( P(x, y) \), we find the new point's x-coordinate by simply adding 4 to the original x-value. Hence, the transformation of the x-coordinate from 3 to 7 is achieved in our exercise.

A vertical stretch in function transformations affects the height of the graph, altering the distance of points from the x-axis. Instead of moving sideways like the horizontal shift, a vertical stretch multiplies the function's output, making the graph taller or shorter depending on the stretch factor.

To comprehensively understand this, consider a transformed function:\[ y = a \cdot f(x) \].

The parameter \( a \) represents the vertical stretch factor. When \( a > 1 \), the graph of the function is stretched away from the x-axis, resulting in it appearing taller. Whereas if \( 0 < a < 1 \), the graph is compressed closer to the x-axis.

• If \( a = 1 \), there is no vertical transformation; everything remains unchanged.
• Here in our exercise, \( y = 2f(x-4) \), \( a = 2 \) suggests each y-coordinate is multiplied by 2.

This doubles the height or depth of each point relative to the x-axis. Initially, at point \( (3, -2) \), the y-value of \( -2 \) becomes \(-4\) after the vertical stretch by 2.

The vertical shift modifies the position of a graph by moving it up or down along the y-axis. Unlike vertical stretches that alter the graph's height, vertical shifts maintain the graph's shape and just translate it vertically.

You can identify a vertical shift from a function in the form:\[ y = f(x) + d \].

Here, the value \( d \) controls the direction and magnitude of the shift. A positive \( d \) value indicates an upward shift, while a negative \( d \) signifies a downward shift. In this exercise, the transformation \( y = 2f(x-4) + 1 \) includes a vertical shift upwards by 1 unit due to \( +1 \).

• This means every point on the graph moves one unit upwards.
• After applying the vertical stretch to the y-coordinate, which results in \(-4\), we add 1 to achieve the new value of \(-3\).

Thus, this step completes the transformation of the point's y-coordinate as part of the function's vertical modifications.