The function y=x^2 has been transformed to y=a f(b x) ....

The function $y=x^{2}$ has been transformed to $y=a f(b x) .$ Determine the equation of each transformed function. a) Its graph is stretched horizontally about the $y$ -axis by a factor of 2 and then reflected in the $x$ -axis. b) Its graph is stretched horizontally about the $y$ -axis by a factor of $\frac{1}{4},$ reflected in the $y$ -axis, and then stretched vertically about the $x$ -axis by a factor of $\frac{1}{4}$.

The function $y=x^{2}$ has been transformed to $y=a f(b x) .$ Determine the equation of each transformed function.
a) Its graph is stretched horizontally about the $y$ -axis by a factor of 2 and then reflected in the $x$ -axis.
b) Its graph is stretched horizontally about the $y$ -axis by a factor of $\frac{1}{4},$ reflected in the $y$ -axis, and then stretched vertically about the $x$ -axis by a factor of $\frac{1}{4}$.

Key Concepts

Function Transformations

Function transformations involve altering the graph of a given function through translations, stretches, compressions, and reflections. This concept is essential for understanding how algebraic modifications to a function’s formula correspond to geometric changes in its graph.

Horizontal Stretch and Compression

Horizontal stretches and compressions modify the x-coordinates of a function’s graph. A stretch about the y-axis by a certain factor involves multiplying the x-variable by a reciprocal factor, effectively widening or narrowing the graph. This concept is crucial when analyzing how the graph scales along the horizontal direction.

Vertical Stretch and Compression

Vertical stretches and compressions affect the y-coordinates of a function’s graph. Multiplying the function by a constant scales the graph vertically, making it taller or shorter. This is a key concept in understanding how the amplitude or vertical scale of the graph changes with different multipliers.

Reflections

Reflections in the coordinate axes involve changing the sign of the input or output variables. Reflecting in the x-axis is achieved by multiplying the entire function by -1, which flips the graph over the x-axis, while reflecting in the y-axis involves replacing the variable x with -x, which mirrors the graph across the y-axis. This concept is important when determining how the orientation of a function’s graph is modified.

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