Suppose that the graph of f(x)=sinx is compressed...

Suppose that the graph of $f(x)=\sin x$ is compressed horizontally by a factor of 2 and then shifted 5 units to the right. $$ \begin{array}{l}{\text { (a) What is the equation for the new graph? }} \\ {\text { (b) What is the equation if you first shift by } 5 \text { and then compress by } 2 ?} \\ {\text { (c) } \mathrm{GU} \text { Verify your answers by plotting your equations. }}\end{array} $$

Suppose that the graph of $f(x)=\sin x$ is compressed horizontally by a factor of 2 and then shifted 5 units to the right.

$$
\begin{array}{l}{\text { (a) What is the equation for the new graph? }} \\ {\text { (b) What is the equation if you first shift by } 5 \text { and then compress by } 2 ?} \\ {\text { (c) } \mathrm{GU} \text { Verify your answers by plotting your equations. }}\end{array}
$$

Key Concepts

Horizontal Compression

Horizontal compression is a transformation that shrinks the graph of a function toward the vertical axis by a given factor. In algebraic terms, replacing x with a multiple (such as 2x) inside the function compresses the graph horizontally by the reciprocal of that multiple, meaning the graph will complete its cycles or patterns in a shorter interval compared to the original function.

Horizontal Translation

Horizontal translation is the shifting of a function's graph left or right without changing its shape. This is achieved by adding or subtracting a constant from the input variable (x), where substituting x with (x - c) shifts the graph to the right by c units and (x + c) shifts it to the left by c units.

Order of Operations in Function Transformations

The order in which transformations are applied can affect the final equation of a function. When multiple transformations like scaling and translation are involved, performing them in different orders may result in different algebraic expressions, even though they visually alter the graph in related ways. Understanding the proper sequence—whether to scale first then translate or vice versa—is critical for correctly determining the transformed function.

Composition of Functions

The composition of functions involves applying one function to the result of another. In the context of transformations, this means substituting one transformation into another by replacing the independent variable appropriately. This concept is essential for combining effects like horizontal scaling and translation into one final composite function expression.

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