Write a formula for the function that results when the given...

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

Key Concepts

Horizontal Translation

Horizontal translation, or shifting, moves the graph of a function left or right along the x-axis. This transformation is implemented by replacing x with (x – h), where h is the distance of the shift. A positive value of h shifts the graph to the right, while a negative value shifts it to the left.

Function Transformations

Function transformations involve modifying a base function through various operations such as scaling, shifting, reflecting, or combining these changes. These operations alter the graph of the function without changing its basic shape, and understanding them is essential for analyzing how the behavior of the function is modified under different conditions.

Horizontal Compression

Horizontal compression is a type of function transformation where the x-coordinates of every point on the graph are multiplied by a factor greater than 1, effectively bringing points closer to the y-axis. This is achieved by replacing x with a multiple of x in the function's formula, thereby reducing the graph's width by the inverse of the scaling factor.

Vertical Translation

Vertical translation shifts the graph of a function upward or downward along the y-axis. This is accomplished by adding or subtracting a constant from the function. For instance, adding a constant moves the graph upward, while subtracting moves it downward, without affecting the overall shape of the graph.

Transcript

Please give Ace some feedback