For the following exercises, write a formula for the...

For the following exercises, write a formula for the function $g$ that results when the graph of a given toolkit function is transformed as described.
The graph of $f(x)=x^{2}$ is vertically compressed by a factor of $\frac{1}{2},$ then shifted to the right 5 units and up 1 unit.

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Key Concepts

Horizontal and Vertical Shifts

Horizontal and vertical shifts move the entire graph without altering its shape. A horizontal shift, achieved by adjusting the variable within the function, moves the graph left or right, while a vertical shift, accomplished by adding or subtracting a constant outside the function, moves the graph up or down. These translations allow the function's position to be adjusted on the coordinate plane without changing its intrinsic nature.

Function Transformations

This concept addresses how the graph of a function can be modified through various operations such as scaling, shifting, reflecting, and rotating. It is fundamental for understanding how changes to the equation of a function directly affect its visual representation on the coordinate plane.

Vertical Compression

Vertical compression refers to scaling the output values of a function by a factor less than 1, which effectively makes the graph appear less steep and more 'compressed' in the vertical direction. This transformation reduces the distance between points on the graph and the x-axis, making the function's peaks and valleys appear closer together.

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