Question
Parent functions QuickChek
Answers
GPT-4o mini
It seems that you're asking about parent functions in mathematics, possibly in the context of a quick review ("QuickChek") format. Parent functions are the simplest forms of functions in a family of functions, and they serve as the foundational building blocks for more complex functions. Below is a summary of some common parent functions along with their general equations, graphs, and characteristics:
### Common Parent Functions:
1. **Constant Function**
- **Equation**: \( f(x) = c \) (where \( c \) is a constant)
- **Graph**: Horizontal line
- **Domain**: All real numbers
- **Range**: \( \{c\} \)
2. **Linear Function**
- **Equation**: \( f(x) = mx + b \) (where \( m \) is the slope, and \( b \) is the y-intercept)
- **Graph**: Straight line
- **Domain**: All real numbers
- **Range**: All real numbers
3. **Quadratic Function**
- **Equation**: \( f(x) = x^2 \)
- **Graph**: Parabola opening upward
- **Domain**: All real numbers
- **Range**: \( [0, \infty) \)
4. **Cubic Function**
- **Equation**: \( f(x) = x^3 \)
- **Graph**: S-shaped curve
- **Domain**: All real numbers
- **Range**: All real numbers
5. **Absolute Value Function**
- **Equation**: \( f(x) = |x| \)
- **Graph**: V-shaped graph opening upwards
- **Domain**: All real numbers
- **Range**: \( [0, \infty) \)
6. **Square Root Function**
- **Equation**: \( f(x) = \sqrt{x} \)
- **Graph**: Half-parabola opening to the right
- **Domain**: \( [0, \infty) \)
- **Range**: \( [0, \infty) \)
7. **Exponential Function**
- **Equation**: \( f(x) = a^x \) (where \( a > 0 \))
- **Graph**: Rapid growth curve
- **Domain**: All real numbers
- **Range**: \( (0, \infty) \)
8. **Logarithmic Function**
- **Equation**: \( f(x) = \log_a(x) \) (where \( a > 1 \))
- **Graph**: Curve that increases slowly
- **Domain**: \( (0, \infty) \)
- **Range**: All real numbers
9. **Rational Function**
- **Equation**: \( f(x) = \frac{1}{x} \)
- **Graph**: Hyperbola
- **Domain**: All real numbers except \( x = 0 \)
- **Range**: All real numbers except \( y = 0 \)
These parent functions serve as essential references for understanding transformations, combinations, and other operations that can be applied to functions. Each has specific characteristics regarding its graph, domain, and range that help in identifying their behavior.
### Common Parent Functions:
1. **Constant Function**
- **Equation**: \( f(x) = c \) (where \( c \) is a constant)
- **Graph**: Horizontal line
- **Domain**: All real numbers
- **Range**: \( \{c\} \)
2. **Linear Function**
- **Equation**: \( f(x) = mx + b \) (where \( m \) is the slope, and \( b \) is the y-intercept)
- **Graph**: Straight line
- **Domain**: All real numbers
- **Range**: All real numbers
3. **Quadratic Function**
- **Equation**: \( f(x) = x^2 \)
- **Graph**: Parabola opening upward
- **Domain**: All real numbers
- **Range**: \( [0, \infty) \)
4. **Cubic Function**
- **Equation**: \( f(x) = x^3 \)
- **Graph**: S-shaped curve
- **Domain**: All real numbers
- **Range**: All real numbers
5. **Absolute Value Function**
- **Equation**: \( f(x) = |x| \)
- **Graph**: V-shaped graph opening upwards
- **Domain**: All real numbers
- **Range**: \( [0, \infty) \)
6. **Square Root Function**
- **Equation**: \( f(x) = \sqrt{x} \)
- **Graph**: Half-parabola opening to the right
- **Domain**: \( [0, \infty) \)
- **Range**: \( [0, \infty) \)
7. **Exponential Function**
- **Equation**: \( f(x) = a^x \) (where \( a > 0 \))
- **Graph**: Rapid growth curve
- **Domain**: All real numbers
- **Range**: \( (0, \infty) \)
8. **Logarithmic Function**
- **Equation**: \( f(x) = \log_a(x) \) (where \( a > 1 \))
- **Graph**: Curve that increases slowly
- **Domain**: \( (0, \infty) \)
- **Range**: All real numbers
9. **Rational Function**
- **Equation**: \( f(x) = \frac{1}{x} \)
- **Graph**: Hyperbola
- **Domain**: All real numbers except \( x = 0 \)
- **Range**: All real numbers except \( y = 0 \)
These parent functions serve as essential references for understanding transformations, combinations, and other operations that can be applied to functions. Each has specific characteristics regarding its graph, domain, and range that help in identifying their behavior.