Let's match the characteristics with their parent functions:
-
Its graph passes through the origin
- Linear (e.g., y = mx + b, where b = 0)
-
Its graph does not touch the x-axis
- Exponential (e.g., y = e^x or any positive base), as the function never crosses the x-axis.
-
For all values of x in its domain y increases as x increases
- Exponential (for positive bases) or Linear (if you consider only positive slopes)
-
Its graph never touches Quadrant II or IV
- Exponential (e.g., y = e^x), as it never has negative y-values.
-
Both its domain and range consist of all real numbers
- Cubic (e.g., y = x^3) or Linear (e.g., y = mx + b)
-
Its graph never touches the y-axis
- Rational (e.g., y = 1/x), as it has a vertical asymptote at x = 0.
-
Its graph is symmetric about the y-axis its range values are all non-negative
- Absolute value (e.g., y = |x|).
-
For all values of x, y values increase at a constant rate
- Linear (e.g., y = mx + b) with a positive slope.
-
Its graph is symmetric about the x-axis
- Odd function (e.g., y = x^3, which is not symmetric about the x-axis but reflects through it).
-
Its domain values are all non-negative
- Square Root (e.g., y = √x).
- Its graph is not continuous
- Rational (e.g., y = 1/x), which has a discontinuity at x = 0.
- Its domain values and range values are non-negative
- Absolute value (e.g., y = |x|) or Square Root (e.g., y = √x).
- Its range values are negative when domain values are negative
- Quadratic (e.g., y = -x^2), which opens downwards and has negative range when x is negative.
- Its domain and range values consist of all real numbers except zero
- Rational (e.g., y = 1/x), which cannot accept x = 0.
To summarize:
- Linear: 1, 3, 5, 8
- Absolute Value: 7, 12
- Quadratic: 13
- Square Root: 10
- Cubic: 5
- Cube Root: None
- Rational: 6, 11, 14
- Exponential: 2, 3, 4
- Logarithmic: None
- None: 9 (as a specific symmetrical function doesn't apply to listed families).