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When we talk about a horizontal shift in graph transformations, we're referring to the movement of the graph along the x-axis. This means that each point on the graph will move left or right by a certain number of units. In the exercise, the transformation from \( f(x) \) to \( f(x + 1) \) involves a horizontal shift to the left by 1 unit.

A positive value added to \( x \) inside the function, like \( f(x + 1) \), indicates a shift to the left. Conversely, subtracting a value, such as in \( f(x - 1) \), results in a shift to the right.

Here's a simple checklist about horizontal shifts:

• \( f(x + c) \): Shift left by \( c \) units
• \( f(x - c) \): Shift right by \( c \) units

By recognizing these patterns, you can quickly understand how the function's graph will be moved along the x-axis.

A vertical stretch occurs when a function's graph is pulled away from the x-axis, making it taller. This transformation involves multiplying the function by a factor larger than 1. In our exercise, we see this transformation in the form of \( 2f(x+1) \).

This means every value of \( f(x+1) \) is scaled by a factor of 2, effectively stretching the graph vertically. This in turn makes each \( y \)-coordinate of a point on the graph twice as far from the x-axis as it was in the original graph.

Some helpful points about vertical transformations:

• If you multiply by a number greater than 1 (e.g., \( 2f(x) \)), the graph stretches.
• If you multiply by a number between 0 and 1 (e.g., \( 0.5f(x) \)), the graph compresses.

This form of transformation changes only the height of the graph without affecting its shape or position on the x-axis.

Graph transformations involve modifying the position, size, and shape of a graph. In mathematics, they are a powerful tool for understanding functions and modeling data. The main types of transformations include shifting, stretching, compressing, and reflecting.

For the given exercise, the transformations are a combination of a horizontal shift (moving left) and a vertical stretch: the original function \( y = f(x) \) is transformed into \( y = 2f(x + 1) - 1 \). This combines several transformations into one:

• Horizontal shift by 1 unit to the left due to \( f(x+1) \).
• Vertical stretch by a factor of 2 due to \( 2f(x+1) \).
• A vertical shift downward by 1 unit from the \(-1\) outside the function.

Each transformation builds upon the last, altering the graph in a precise way to produce new coordinates.

Coordinates are essential in understanding any graph transformation, as they define the position of points on the graph. When you transform a graph, each point, represented as \((x, y)\), moves to a new location based on the rules applied.

For instance, in the exercise, the point \((a, b)\) on the original graph of \( y = f(x) \) is transformed to the point \((a-1, 2b-1)\) on the new graph of \( y = 2f(x+1) - 1 \). Let's break this down:

• Horizontal transformation: Shift the x-coordinate by subtracting 1, giving \(a-1\).
• Vertical transformation: Multiply the y-coordinate by 2 and then subtract 1 to get \(2b-1\).

By properly adjusting the coordinates based on the type of transformation (horizontal or vertical), you can accurately determine the position of a point on any transformed graph.