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GPS Algebra Review

GPS Algebra Review. All standards. GPS Algebra. Unit 1: Chances of Winning Unit 2: Unit 3: Unit 4: Unit 5: Unit 6:. Unit 5 pt 1. MM2A3. Students will analyze quadratic functions in the forms f ( x ) = ax 2 + bx + c and f ( x ) = a ( x – h ) 2 + k .

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GPS Algebra Review

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  1. GPS Algebra Review All standards

  2. GPS Algebra • Unit 1: Chances of Winning • Unit 2: • Unit 3: • Unit 4: • Unit 5: • Unit 6:

  3. Unit 5 pt 1 MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k. • Convert between standard and vertex form. • Graph quadratic functions as transformations of the function f(x) = x2. • Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. • Explore arithmetic series and various ways of computing their sums. MM2N1. Students will represent and operate with complex numbers. • Write square roots of negative numbers in imaginary form. • Write complex numbers in the form a + bi. • Add, subtract, multiply, and divide complex numbers. • Simplify expressions involving complex numbers. MM2A4. Students will solve quadratic equations and inequalities in one variable. • Solve equations graphically using appropriate technology. • Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. • Analyze the nature of roots using technology and using the discriminant. • Solve quadratic inequalities both graphically and algebraically, and describe the solutions using linear inequalities.

  4. Unit 5 pt 2 • MM1A3.a: a. Solve quadratic equations in the form ax2+ bx + c = 0, where a = 1, by using factorization and finding square roots where applicable. • MM1A3.c: c. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of x2+ bx + c = 0. • MM1A3.d: d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

  5. MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k.a. Convert between standard and vertex form.b. Graph quadratic functions as transformations of the function f(x) = x2.c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change.d. Explore arithmetic series and various ways of computing their sums. Convert between standard and vertex form 2x2 + 4x + 4 (2x2 + 4x)+ 4 =2(x2 + 2x) + 4 2(x2 + 2x + 1)+ 4 – 2(1) From Vertex: From Standard: Recall: y = (x+2)2 – 1 = (x+2)(x+2) – 1 = x2 + 2x + 2x + 4 – 1 = x2 + 4x + 3 Group x terms Rewrite using Binomial Theorem *a should always equal 1* • Find (b/2)2 • Add inside () and • subtract outside () FOIL Gather like terms ( )2 b 2 Factor and Simplify 2(x+1)(x+1)+ 4 – 2 = 2(x+1)2 + 2 Standard Form Vertex Form

  6. MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k.a. Convert between standard and vertex form.b. Graph quadratic functions as transformations of the function f(x) = x2.c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change. Graph quadratic functions as transformations of the function f(x) = x2. From Vertex: Next: Solve for those two points AND Graph Step 1: Find vertex Step 2: Pick 2 x points (one before vertex and one after) From Standard: Step 1: Find axis of symmetry Let’s GRAPH!  Step 2: Plug in axis value to find vertex Step 3: Pick 2 x points (one before vertex and one after)

  7. MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2+ bx+ c and f(x) = a(x – h)2+ k.a. Convert between standard and vertex form.b. Graph quadratic functions as transformations of the function f(x) = x2.c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change.

  8. Unit 4 Recall 1. List all transformations. y = -2(x+2)3 – 4 2. Label. Even, Odd or Neither a. b. c. y = x3 + x d. x4 + 2x2 + 1 3. Solve the following. a. √ 7 + 3x = 5 b. 1 = 2 3x 5x2

  9. Unit 4 Standards • MM1A1.c: c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. • MM1A1.d: d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. • MM1A1.h: h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither.

  10. MM1A1.c: c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes. y = - a (-x – h)n + k Horizontal shift

  11. MM1A1.d: d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.

  12. MM1A1.d: d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.

  13. MM1A1.d: d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior.

  14. MM1A1.h: h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither.

  15. MM1A3.a: a. Solve quadratic equations in the form ax2+ bx + c = 0, where a = 1, by using factorization and finding square roots where applicable.

  16. MM1A3.c: c. Use a variety of techniques, including technology, tables, and graphs to solve equations resulting from the investigation of x2+ bx + c = 0.

  17. MM1A3.d: d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

  18. Unit 3 Recall • List all the parent functions. Include a) name, b) equation and c) graph • 2, 4, 6, 8,… • What is the 5th term? • What is the 12th term? • Write the closed and recursive formula. • Find the rate of change.` Recall: a1 = first term an = d(n – 1) + a1 an = an-1 + d (Recusive) (Closed)

  19. Unit 3 Standards • MM1A1a.: Represent functions using function notation. • MM1A1.b.: Graph the basic functions f(x) = xn , where n = 1 to 3, f(x) = √x , f(x) = |x|, and f(x) = 1 \x. • MM1A1.d.: Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior. • MM1A1.e.: Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. • MM1A1.f.: Recognize sequences as functions with domains that are whole numbers. • MM1A1.g.: Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families.

  20. y = ax2 + bx + c f(x) = ax2 + bx + c • MM1A1a.: Represent functions using function notation Also…. Given a graph: Find f(2) Also…. Given an equation: f(x) = x2 + 4x – 9 Find f(2) Remember this is y f(x) = x2 + 4x – 9 f(x) = (2) 2 + 4(2) – 9 f(x) = 4 + 8 – 9 f(x) = 3 f(2) Find the value of y when x = 2 Therefore f(2) = 0

  21. y = ax2 + bx + c f(x) = ax2 + bx + c • MM1A1a.: Represent functions using function notation Also…. Given a graph: Find f(-1) Also…. Given an equation: f(x) = x2 + 4x – 9 Find f(2) Remember this is y f(x) = x2 + 4x – 9 f(x) = (2) 2 + 4(2) – 9 f(x) = 4 + 8 – 9 f(x) = 3 f(-1) Find the value of y when x = -1 Therefore f(-1) = 6

  22. MM1A1.b.: Graph the basic functions f(x) = xn , where n = 1 to 3, f(x) = √x , f(x) = |x|, and f(x) = 1 \xMM1A1.e.: Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior. Parent Functions Parent Functions: You need to know: graphs names tables (ex) characteristics Reference FOLDABLES

  23. MM1A1.g.: Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. Constant Variable Example: Find the rate of change of f(2) and f(-1) when f(x) = x2 + 4x – 9 Recall: EQUATION

  24. MM1A1.g.: Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of change of linear, quadratic, square root, and other function families. Example: Find the rate of change of f(2) and f(-1) when f(x) = x2 + 4x – 9 We found f(2) = 3 You need f(-1) f(x1) – f(x2) x1 – x2 f(2) – f(-1) 2 – (-1) = f(x) = x2 + 4x – 9 f(-1) = (-1) 2 + 4(-1) – 9 = 1+-4– 9 = -12 (3) – (-12) 2 – (-1) = 15 3 = Now plug it into the equation. = 5

  25. MM1A1.f.: Recognize sequences as functions with domains that are whole numbers. Recall: a1 = first term an = d(n – 1) + a1 an = an-1 + d (Recusive) (Closed) You only change the first term and the difference d a1 Study the equations!

  26. Unit 2 Recall 1.

  27. Unit 2 Standards • MM1A2.a.: Simplify algebraic and numeric expressions involving square root. • MM1A2.b.: Perform operations with square roots. • MM1A2.c.: Add, subtract, multiply, and divide polynomials. • MM1A2.d.: Expand binomials using the Binomial Theorem. • MM1A2.e.: Add, subtract, multiply, and divide rational expressions. • MM1A2.g: Use area and volume models for polynomial arithmetic. Independent Practice

  28. Unit 1 Standard • MM1D3.a: Compare summary statistics (mean, median, quartiles, and interquartile range) from one sample data distribution to another sample data distribution in describing center and variability of the data distributions. • MM1D3.b: Compare the averages of the summary statistics from a large number of samples to the corresponding population parameters. c: Understand that a random sample is used to improve the chance of selecting a representative sample. • MM1D2. a. Find the probabilities of mutually exclusive events. b. Find the probabilities of dependent events. c. Calculate conditional probabilities. d. Use expected value to predict outcomes. • MM1A2.d: d. Expand binomials using the Binomial Theorem. • MM1D1 a. Apply the addition and multiplication principles of counting. b. Calculate and use simple permutations and combinations.

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