systems and quadratics
DESCRIPTION
Algebra 2 unit of solving systems of equations, including linear-quadratic systemsTRANSCRIPT
Algebra 2 Systems of Equations 1
Classwork: Composition of Functions, Solving a System by elimination Given 𝑓 𝑥 = 2𝑥 + 1 and 𝑔(𝑥) = 𝑥!, find:
1. 𝑓 𝑔 3
2. g(f(1))
3. 𝑓 𝑔 𝑥 =
4. 𝑔(𝑓(𝑥))=
5. 𝑓(𝑓(𝑥))
Example 1 A system with one solution.
3𝑥 − 8𝑦 = 21−5𝑥 + 6𝑦 = 9
Multiply the first equation by 5, and the second one by 3. This creates coefficients of x that are inverses; 15x and -‐15x. Those terms will cancel when added.
15𝑥 − 40𝑦 = 105−15𝑥 + 18𝑦 = 27
Add the two equations by combing the like terms.
0𝑥 − 22𝑦 = 132 −22𝑦 = 132 𝑦 = −6
Solve for y.
3𝑥 − 8 −6 = 21 3𝑥 = 69 𝑥 = 13
Find x using one of the original equations
The solution is the point (13, -‐6).
Express your answer an as ordered pair. The graphs intersect at the single point (13, -‐6).
Algebra 2 Systems of Equations 2
Homework: Solving a System of Linear Equations Solve each system by substitution.
1. 2x + 3y = 5 2. 4x + 6y = 15 x – 5y = 9 -‐x + 2y = 5
Solve each system by eliminating one variable, using linear combination. 3. 3x + 4y = -‐4 4. 3x + 2y = 6 x + 2y = 2 6x + 3y = 6 5. 2x – 5y = 10 6. 4x – 3y = 0 -‐3x + 4y = -‐15 10x – 7y = 2 7. This system of equations does NOT have exactly one solution. What sort of solutions does it have? Solve and interpret your result. 4x + 6y = -‐11 2x + 3y = -‐ 1
Algebra 2 Systems of Equations 3
Classwork: Composition of Functions, Solving a System of Equations Bell-‐Work: Composition of Functions Given f(𝑥) = 2𝑥 + 5 and 𝑔(𝑥) = 3𝑥!, find:
1. 𝑓(𝑔(1))
2. 𝑔(𝑓(−4))
3. 𝑓(𝑓(5))
4. 𝑓(𝑔(𝑥))
5. 𝑔(𝑓(𝑥))
6. 𝑓(𝑓(𝑥))
Systems of equations with one, none, or infinite solutions
Notes on solutions
7. Solve the System 2𝑥 − 4𝑦 = 134𝑥 − 5𝑦 = 8
8. Solve the System 𝑥 − 2𝑦 = 32𝑥 − 4𝑦 = 7
Algebra 2 Systems of Equations 4
Notes
9. Solve the system 6𝑥 − 10𝑦 = 12
−15𝑥 − 25𝑦 = −30
Linear-‐Quadratic Systems of Equations
10. Solve the system 𝑦 = 𝑥 𝑦 = 𝑥!
Algebra 2 Systems of Equations 5
Classwork: Composition and Inverses of Functions; Solving Systems Composition of Functions 1. Given 𝑓(𝑥) = 3𝑥 + 2 and 𝑔(𝑥) = 𝑥−23 , find 𝑓(𝑔(𝑥)). 2. Given 𝑓(𝑥) = (𝑥 + 1)! and 𝑔 𝑥 = 𝑥3 − 1, find 𝑓(𝑔(𝑥)). Definition of inverse.
If f(g(x)) = x and g(f(x)) = x, then f and g are inverses. 3. Are these functions inverses of each other? 𝑓(𝑥) = (𝑥 + 1)! 𝑔 𝑥 = 𝑥! + 1
• Test with a value. Find f(g(a)) for some value x = a.
• Test with composition. Find f(g(x)). Solving a Linear-‐Quadratic system of equations
4. Solve the system, using linear combination. 𝑦 = 𝑥! − 5𝑥 + 7𝑦 = 2𝑥 + 1
5. Solve the system, using linear combination. 𝑦 = 𝑥! + 4𝑦 = 𝑥 + 1
Algebra 2 Systems of Equations 6
Classwork: Solving Systems
1. Describe, using an example, a procedure for solving a system of two linear equations. Explain each step, and the rationale for it.
2. Given a system of two linear equations, what is the geometric meaning of there being:
a. Exactly one solution
b. No solution
c. An infinite number of solutions
3. Given a system of two linear equations, what is the algebraic meaning of there being:
a. Exactly one solution
b. No solution
c. An infinite number of solutions
4. Given a system of one linear equation and one quadratic equation, describe all the possible numbers of solutions.
Practice for Unit Test Inverses
1. Find the inverse of 𝑦 = !!𝑥 + 8.
2. Find the inverse of 𝑦 = 2𝑥 + 8 !. Use composition to prove that your inverse is correct.
Algebra 2 Systems of Equations 7
Solving a Quadratic
3. Solve the equation for x. 𝑥! − 12𝑥 − 28 = 0
4. Solve for x: 3𝑥! + 10𝑥 + 3 = 0
5. Solve for x: 3𝑥! − 6𝑥 + 12 = 0
6. Solve for x: 𝑥! + 10𝑥 − 3 = 0
7. Solve for x 7𝑥! − 2𝑥 − 9 = 0 Solving a System of Equations, using Elimination
8. Solve the system 2𝑥 − 4𝑦 = 13 4𝑥 − 5𝑦 = 8
9. Solve the system
7𝑥 − 12𝑦 = −22 −5𝑥 + 8𝑦 = 14
10. Solve the system −9𝑥 + 6𝑦 = 0 −12𝑥 + 8𝑦 = 0
11. Solve the system 6𝑥 + 9 𝑦 = −3 −4𝑥 − 6𝑦 = 11
12. Solve the system, using elimination 𝑦 = 2𝑥 + 1 𝑦 = 𝑥! − 2
13. Solve the system 3𝑥 − 2𝑦 + 10 = 0 2𝑥! + 12𝑦 + 13 = 0
14. A rocket is launched from the ground and follows a parabolic path described by the equation y = -‐
x2 + 10x . At the same time, a flare is launched from 10 feet above ground and follows a straight path described by the equation y = -‐x + 10. Solve this system of equations to find the coordinates where their paths intersect.
Algebra 2 Systems of Equations 8
Test-‐A: Systems of Equations & Composition Name: ________________________________
• Partial credit is given for evidence of correct, but incomplete, work. • Unsupported answers will not receive credit.
1. Janina is solving a system of equations, and gets 0 = 19 as the result. What does this tell her about these equations, and what should she write down as her answer for the solution of the system? 2. Write a simple system of equations that will have NO solution. 3. Solve the system, using substitution. Write your solution as an ordered pair, (x, y).
𝑥 + 6𝑦 = 1 2𝑥 + 11𝑦 = 4
4. Solve the system, using elimination (linear combination). Write your solution as an ordered pair, (x, y).
2𝑥 + 3𝑦 = 55𝑥 + 7𝑦 = 8
Algebra 2 Systems of Equations 9
5. Solve the system, using elimination (linear combination). Give your answer as an ordered pair. 4𝑥 + 𝑦 = 𝑥! − 2
𝑥 − 𝑦 = 2
6. Given 𝑓 𝑥 = 2𝑥! + 5 and 𝑔 𝑥 = 𝑥3
2 − 5, determine whether or not they are inverses, and verify your answer.
7. Solve the equation for x, using the Quadratic Formula: 𝑥 = −𝑏± 𝑏2−4𝑎𝑐2𝑎
𝑥! − 8𝑥 + 20 = 0
Algebra 2 Systems of Equations 10
TEST-‐B: Systems of Equations & Composition Name: ________________________________
• Partial credit is given for evidence of correct, but incomplete, work. • Unsupported answers will not receive credit.
1. Jamila is solving a system of equations, and gets 0 = 0 as the result. What does this tell her about these equations, and what should she write down as her answer for the solution of the system? 2. Write a simple system of equations that will have an INFINTE number of solutions. 3. Solve the system, using substitution. Write your solution as an ordered pair, (x, y).
7𝑥 + 𝑦 = 6 𝑥 − 2𝑦 = −12
4. Solve the system, using elimination (linear combination). Write your solution as an ordered pair, (x, y).
7𝑥 − 2𝑦 = −95𝑥 − 3𝑦 = 3
Algebra 2 Systems of Equations 11
5. Solve the system, using elimination (linear combination). Give your answer as an ordered pair. 𝑦 − 4𝑥 = 𝑥! + 5
𝑦 − 𝑥 = 5
6. Given 𝑓 𝑥 = 2𝑥! + 5 and 𝑔 𝑥 = 𝑥−52
3 , determine whether or not they are inverses, and verify your answer.
7. Solve the equation for x, using the Quadratic Formula: 𝑥 = −𝑏± 𝑏2−4𝑎𝑐2𝑎
𝑥! − 16𝑥 + 4 = 0