List the transformations that take
g(x)=1/x to f(x)=3x-4/x+2.
Is it
1. reflection across y axis?
2. translate left 2 units?
3. translate up 3 units?
4. vertically stretch by a factor of 10?
Assuming the usual sloppiness with parentheses, I'll assume that
f(x) = (3x-4)/(x+2) = -10/(x+2) + 3
so you appear to be correct, with the caveat that the stretch is done before the upward shift.
To determine the transformations that take g(x) = 1/x to f(x) = (3x - 4)/(x + 2), we need to compare the two functions and identify the changes that have occurred. Let's examine each transformation one by one.
1. Reflection across the y-axis:
To reflect a function across the y-axis, the sign of the x-term would change. In this case, we see that the sign of the x-term has not changed from g(x) to f(x), so there is no reflection across the y-axis.
2. Translation left 2 units:
To translate a function left by a certain number of units, the entire function needs to be shifted horizontally. Notice that g(x) = 1/x is not changed in terms of x, while f(x) = (3x - 4)/(x + 2) has x + 2 in the denominator. This indicates that there has been a horizontal shift to the left by 2 units. Therefore, f(x) is a translation of g(x) left 2 units.
3. Translation up 3 units:
To translate a function up by a certain number of units, the entire function needs to be shifted vertically. Looking at g(x) and f(x), there is no indication of a change in the y-values. Therefore, f(x) is not a translation up 3 units.
4. Vertical stretch by a factor of 10:
To vertically stretch a function, the y-values are multiplied by a certain factor. From g(x) = 1/x to f(x) = (3x - 4)/(x + 2), we don't observe a consistent factor applied to the y-values. Therefore, there is no vertical stretch by a factor of 10.
In conclusion, the transformations that take g(x) = 1/x to f(x) = (3x - 4)/(x + 2) are a translation left 2 units.