Linear Functions and Systems — Algebra 2 Unit 2 practice.
This unit covers function notation, systems of three variables and linear programming — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers function notation, systems of three variables and linear programming — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Function Notation
Function notation f(x) means the output of function f when the input is x. Students must be able to evaluate functions, interpret f(a) = b, and perform operations like f(x) + g(x) or f(g(x)). Exams test substitution, reading graphs, and understanding domain and range.
Key Points
- f(3) means substitute x = 3 into the function and simplify
- f(x) = y means x is the input (domain) and y is the output (range)
- Composite function f(g(x)): evaluate the inner function first, then plug that result into the outer function
- A relation is a function only if every x-value maps to exactly one y-value (vertical line test)
Given f(x) = 2x − 5 and g(x) = x², find f(g(3)).
First evaluate the inner function: g(3) = 3² = 9. Then substitute that result into f: f(9) = 2(9) − 5 = 18 − 5 = 13. So f(g(3)) = 13.
2 Systems Of Three Variables
A system of three equations with three variables (x, y, z) is solved using elimination or substitution to reduce the system step by step until one variable is isolated. Students must know how to back-substitute to find all three values and check the solution in all original equations. Exams typically require setting up and solving the full system from a word problem or given equations.
Key Points
- Pick any two equation pairs and eliminate the same variable from each pair — this gives a 2×2 system
- Solve the resulting 2×2 system for two variables, then back-substitute to find the third
- A solution (x, y, z) must satisfy ALL three equations simultaneously
- If elimination produces a contradiction (e.g., 0 = 5), there is no solution; if it produces 0 = 0, there are infinitely many
Solve: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 4.
Add equations 1 and 2 to eliminate y: 3x + 2z = 9. Add equations 1 and 3 to eliminate z: 2x + 3y = 10 — wait, instead add eq 1 and eq 3 to get 2x + 3y = 10, giving a 2×2 system. Solve that system to get x = 1, y = 2, then substitute back into equation 1 to find z = 3. The solution is (1, 2, 3).
3 Linear Programming
Linear programming finds the maximum or minimum value of an objective function subject to a set of linear inequality constraints. Students must graph the feasible region (the area satisfying all constraints), identify the corner points (vertices), and test each vertex in the objective function. Exams require setting up the system from a word problem, graphing correctly, and stating which vertex gives the optimal value.
Key Points
- The optimal (max or min) value always occurs at a corner point (vertex) of the feasible region
- Graph each constraint as a boundary line, then shade the correct side; the feasible region is the overlap
- Find vertices by solving the pairs of boundary-line equations where they intersect
- Substitute every vertex into the objective function and compare — do not skip any corner point
Maximize P = 3x + 2y subject to: x + y ≤ 4, x ≥ 0, y ≥ 0.
Graph the constraints to find the feasible region is a triangle with vertices at (0, 0), (4, 0), and (0, 4). Evaluate P at each vertex: P(0,0) = 0, P(4,0) = 12, P(0,4) = 8. The maximum value is P = 12 at the point (4, 0).
Questions, answered.
What is Linear Functions and Systems?
Linear Functions and Systems is Unit 2 of Algebra 2, covering function notation, systems of three variables and linear programming.
How to study for Algebra 2 Unit 2?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.