91Ó°ÊÓ

Graph shifts modify the position of the graph of a function in the coordinate plane. An important factor in graph shifts is determining whether the movement is horizontal or vertical. Shifting a graph horizontally means moving it left or right along the x-axis. This is accomplished by altering the input value, \(x\), inside the function. For example, when a function is presented as \(f(x-a)\), the graph is shifted \(a\) units to the right, as seen in the solution for \(y=3f(x-1)\), where the shift is 1 unit to the right.
Vertical shifts, on the other hand, involve moving the graph up or down along the y-axis. This occurs by adjusting the function's output, either increasing or decreasing all y-values by a certain amount. Knowing these types of shifts allows you to visualize how the shape of a graph translates in space.
To summarize graph shifts:

• Horizontal shifts: Change values inside the function; move the graph left or right.
• Vertical shifts: Change the function's final output; move the graph up or down.

Understanding these basic movements is essential to mastering function transformations.

A vertical stretch involves changing the distance between points on the graph and the x-axis. This is done by multiplying the function by a factor greater than one. For example, a graph experiencing a vertical stretch by a factor of \(3\) will have all y-values become three times their original value. This effectively pulls the graph away from the x-axis, making it appear taller and skinnier.
When analyzing transformations, an easy way to identify a vertical stretch is by looking at the coefficient placed before the function \( f \). In the transformation from \( y = f(x) \) to \( y = 3f(x-1) \), the coefficient \(3\) indicates our vertical stretch.
Points to remember about vertical stretch:

• Multiplier greater than 1 stretches the graph away from the x-axis.
• All vertical distances (y-values) increase by the factor of the multiplier.

This transformation does not affect the x-values but solely alters the height of the graph, making each point more pronounced.

The horizontal shift is a subtle yet crucial transformation that affects how a graph sits along the x-axis. This change is caused by modifying the input \(x\) in the function's equation. When a graph shifts horizontally, specific transformations occur based on the adjustments to these inputs. For instance, \(f(x-a)\) results in a shift \(a\) units to the right, as demonstrated by the basic function transformation in the exercise.
For the given example \(y = 3f(x-1)\), the term \(x-1\) inside the function indicates a horizontal shift one unit to the right. This change slightly repositions the function's shape without modifying its overall form.
Key takeaways for horizontal shifts:

• Shift direction depends on the sign: "minus" shifts right, "plus" shifts left.
• Makes no difference to the vertical orientation of the graph.

Recognizing and calculating horizontal shifts is essential in accurately graphing and interpreting functions, providing clarity within transformations.