Here’s a table that matches the characteristics with their corresponding parent functions:
| Characteristics | Parent Functions | |--------------------------------------------------------------|----------------------------------| | Its graph passes through the origin | Linear, Cubic, Square Root, Cube Root | | Its graph does not touch the x-axis | Absolute Value, Exponential, Logarithmic | | For all values of x in its domain y increases as x increases | Linear, Exponential, Absolute Value | | Its graph never touches Quadrant II or IV | Exponential, Absolute Value | | Both its domain and range consist of all real numbers | Linear, Cubic, Quadratic, Cube Root, Rational | | Its graph never touches the y-axis | Rational | | Its graph is symmetric about the y-axis, and its range values are all non-negative | Absolute Value, Quadratic | | For all values of x, y values increase at a constant rate | Linear | | Its graph is symmetric about the x-axis | None (though some may be symmetric in context) | | Its domain values are all non-negative | Absolute Value, Square Root | | Its graph is not continuous | Rational | | Its domain values and range values are non-negative | Absolute Value, Square Root | | Its range values are negative when domain values are negative | Cubic, Rational | | Its domain and range values consist of all real numbers except zero | Rational |
Please note that some characteristics can apply to multiple functions, and certain characteristics can also have nuanced interpretations that might lead to different associations.