To determine how the parent function [tex]\( f(x) \)[/tex] was transformed to make [tex]\( g(x) \)[/tex], we need to consider the different types of transformations that can be applied to the function. Here are the possible transformations:
1. Vertical Shifts:
- Upward Shift: If [tex]\( g(x) = f(x) + k \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is shifted upward by [tex]\( k \)[/tex] units.
- Downward Shift: If [tex]\( g(x) = f(x) - k \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is shifted downward by [tex]\( k \)[/tex] units.
2. Horizontal Shifts:
- Right Shift: If [tex]\( g(x) = f(x - h) \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is shifted to the right by [tex]\( h \)[/tex] units.
- Left Shift: If [tex]\( g(x) = f(x + h) \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is shifted to the left by [tex]\( h \)[/tex] units.
3. Vertical Stretching and Compressing:
- Vertical Stretch: If [tex]\( g(x) = a \cdot f(x) \)[/tex] where [tex]\( |a| > 1 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is stretched vertically by a factor of [tex]\( a \)[/tex].
- Vertical Compression: If [tex]\( g(x) = a \cdot f(x) \)[/tex] where [tex]\( 0 < |a| < 1 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is compressed vertically by a factor of [tex]\( a \)[/tex].
4. Horizontal Stretching and Compressing:
- Horizontal Compression: If [tex]\( g(x) = f(b \cdot x) \)[/tex] where [tex]\( |b| > 1 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a factor of [tex]\( b \)[/tex].
- Horizontal Stretch: If [tex]\( g(x) = f(b \cdot x) \)[/tex] where [tex]\( 0 < |b| < 1 \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is stretched horizontally by a factor of [tex]\( b \)[/tex].
5. Reflections:
- Reflection across the x-axis: If [tex]\( g(x) = -f(x) \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is reflected across the x-axis.
- Reflection across the y-axis: If [tex]\( g(x) = f(-x) \)[/tex], then the graph of [tex]\( f(x) \)[/tex] is reflected across the y-axis.
When analyzing the given functions, we need to compare the specific forms of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to identify which transformations have been applied. Here are the types of transformations you would look for:
- Vertical Shift: Check for addition or subtraction of a constant to [tex]\( f(x) \)[/tex].
- Horizontal Shift: Check for addition or subtraction within the argument of [tex]\( f \)[/tex].
- Vertical Stretch/Compression: Check for multiplication by a constant outside the function.
- Horizontal Stretch/Compression: Check for multiplication within the argument of [tex]\( f \)[/tex].
- Reflections: Check for negation of the function or its argument.
In summary, the transformations that can be applied to the parent function [tex]\( f(x) \)[/tex] to obtain [tex]\( g(x) \)[/tex] are as follows:
1. Vertical Shift
2. Horizontal Shift
3. Vertical Stretch/Compression
4. Horizontal Stretch/Compression
5. Reflection
These categories encompass all the basic transformations that can modify a function.
