Page "Analytic geometry" Paragraph 21

Transformations are applied to parent functions to turn it into a new function with similar characteristics.

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative.

The k and h values introduce translations, h, vertical, and k horizontal.

Positive h and k values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.