State the transformations. Identify holes, vertical asymptotes, and horizontal asymptotes of each. (I'm fine with the asymptotes, just need help with stating the trandformations)
f(x)= (3x+9)/(x+2)
To identify the transformations of the function f(x) = (3x + 9)/(x + 2), we need to understand the properties of rational functions.
1. Rational function: A rational function is a ratio of two polynomials, where the numerator and denominator are both polynomials. In this case, the numerator is 3x + 9, and the denominator is x + 2.
2. Transformations:
a. Translation/Shift: The numerator and denominator of the rational function affect the position of the graph. In this case, the numerator has a horizontal shift of 0 (no shift) since x is not affected by any manipulation. However, the denominator has a horizontal shift of -2 units to the left compared to the standard form (x). This means the graph of the function is shifted two units to the left.
b. Vertical Stretch/Compression: The coefficient in front of x in the numerator and denominator affects the vertical stretch or compression. In this case, the numerator has a coefficient of 3 in front of x, which means there is a vertical stretch by a factor of 3 compared to the standard form (f(x) = x). The denominator does not have a coefficient multiplying x, which is equivalent to a coefficient of 1, meaning there is no vertical stretch or compression.
c. Vertical Translation: The constant term in the numerator and denominator affects the vertical translation. In this case, the numerator has a constant term of 9, which means there is a vertical translation of 9 units up compared to the standard form (f(x) = x). The denominator has a constant term of 2, which can be considered a vertical translation of 2 units down.
Therefore, the transformations of the function f(x) = (3x + 9)/(x + 2) can be summarized as follows:
- Horizontal shift: 2 units to the left
- Vertical stretch: By a factor of 3
- Vertical translation: 9 units up for the numerator and 2 units down for the denominator