Problem Analysis:
The image provided is from a section on "Parent Functions and Transformations" in Algebra 2. The task involves understanding parent functions, their graphs, domains, and ranges. The specific question appears to be related to identifying the
most basic function in each family of functions.
Key Concepts:
1.
Parent Functions: These are the simplest functions in a family of functions. They serve as the foundation for understanding transformations (shifts, stretches, compressions, reflections).
2.
Domain: The set of all possible input values (\(x\)-values) for which the function is defined.
3.
Range: The set of all possible output values (\(y\)-values) that the function can produce.
Parent Functions Listed:
-
Constant Function: \( f(x) = 1 \)
-
Linear Function: \( f(x) = x \)
-
Absolute Value Function: \( f(x) = |x| \)
-
Quadratic Function: \( f(x) = x^2 \)
Graphs and Characteristics:
1.
Constant Function (\( f(x) = 1 \)):
- Graph: A horizontal line at \( y = 1 \).
- Domain: All real numbers (\( \mathbb{R} \)).
- Range: \( \{1\} \) (only the value 1).
2.
Linear Function (\( f(x) = x \)):
- Graph: A straight line passing through the origin with a slope of 1.
- Domain: All real numbers (\( \mathbb{R} \)).
- Range: All real numbers (\( \mathbb{R} \)).
3.
Absolute Value Function (\( f(x) = |x| \)):
- Graph: A V-shaped curve with its vertex at the origin.
- Domain: All real numbers (\( \mathbb{R} \)).
- Range: \( [0, \infty) \) (all non-negative real numbers).
4.
Quadratic Function (\( f(x) = x^2 \)):
- Graph: A parabola opening upwards with its vertex at the origin.
- Domain: All real numbers (\( \mathbb{R} \)).
- Range: \( [0, \infty) \) (all non-negative real numbers).
Solution Explanation:
The question asks to identify the
most basic function in each family. The term "basic" here refers to the simplest form of the function without any transformations (such as shifts, stretches, or reflections).
- For the
constant function, the most basic form is \( f(x) = 1 \).
- For the
linear function, the most basic form is \( f(x) = x \).
- For the
absolute value function, the most basic form is \( f(x) = |x| \).
- For the
quadratic function, the most basic form is \( f(x) = x^2 \).
These are the parent functions listed in the table, and they are indeed the simplest forms of their respective families.
Final Answer:
The most basic function in each family is already given in the table:
- Constant: \( f(x) = 1 \)
- Linear: \( f(x) = x \)
- Absolute Value: \( f(x) = |x| \)
- Quadratic: \( f(x) = x^2 \)
Thus, the solution is:
\[
\boxed{\text{basic}}
\]
Parent Tip: Review the logic above to help your child master the concept of algebra 2 parent functions and transformations worksheet.
