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TOPICS ON THE MIDTERMTHESE TOPICS ARE JUST PULLED FROM THE CHAPTER REVIEW FROM YOUR TEXTBOOK (pages are listed below). THEY ARE A GREAT WAY TO DO A VERY QUICK REVIEW OF EVERY TOPIC AND SEE ONE EXAMPLE PROBLEM FROM EACH SECTION IN YOUR BOOK. Remember that these examples are just one problem from each section and don’t represent all the types of problems from that section, but it does give you a fairly representative example of the key concept from each section.BELOW ARE THE CHAPTER NUMBERS WITH THE PAGE NUMBERS OF THE CHAPTER REVIEW AND CHAPTER TEST (THESE ARE REALLY NICE QUICK OVERVIEWS OF EACH SECTION AND A NICE WAY TO STUDY! PRACTICEING PROBLEMS IS A GREAT WAY TO STUDY!)CHAPTER 1: EQUATIONS AND INEQUALITIES (Review: Page 60-64 Test: Page 65)CHAPTER 2: LINEAR EQUATIONS AND FUNCTIONS (Review: Pg 140-144 Test: Page 145)CHAPTER 3: LINEAR SYSTEMS AND MATRICES (Review: Pg 221-226 Test: Page 227)CHAPTER 4: QUADRATIC FUNCTIONS (Review: Pg 318-322 Test: Page 323)CHAPTER 5.1-5.4: POLYNOMIAL FUNCTIONS (Pg. 401-404 Review, Pg. 407 Test (#1-16)Note: We did the order a bit differently in our class. We started with what functions are and are not (section 2.1) and then did most of chapter 1 before doing the rest of Chapter 2.Chapter 1: EQUATIONS AND INEQUALITIES1.1 Apply the Property that the statement illustrates (Key: Properties include the commutative, distributive, associative etc.)Example: Identify the property that the statement illustrates. 2(w + l) = 2w + 2l Answer: DISTRIBUTIVE PROPERTYNote we skipped this section of the book. Don’t worry about it too much. You use these properties all the time anyways.1.2 Evaluate and Simplify Algebraic Expressions (KEY: Combine like terms)Example: Simplify the expression 5(y – 4) – 3(2y – 9) 1.3 Solve Linear Equations (KEY: ISOLATE THE VARIABLE)Example: Solve -4(3x + 5) = -2(5 – x)1.4 Rewrite Formulas and Equations (KEY: Plug in for the known variable and solve for the unknown.)Example: Solve 5x – 11y = 7 for y. Then find the value of y when x = 4.1.5 use Problem Solving Strategies and Models (KEY: Really focus on what the question is asking for.)Example: Find the time it takes to drive 525 miles at 50 miles per hour.1.6 Solve Linear Inequalities (KEY: Solve equations. Flip the sign if dividing by a negative. Open circle or closed circle depending on the symbol.Example: Solve 35 – 3x < 14. Then graph the solution on a number line.1.7 Solve Absolute Value Equations and Inequalities (KEY: You are solving two equations! Don’t forget to flip the sign when you make the second equation negative!Example: Solve |3x – 7| > 2Chapter 2: LINEAR EQUATIONS AND FUNCTIONSFORMS OF LINEAR FUNCTIONS:Standard Form: Ax + By = CSlope Intercept Form: y = mx + bPoint Slope Form: y – y1= m(x – x1)2.1 Represent Relations and Functions (KEY: Functions have exactly one y for every x term)Example: Tell whether the relation given is a function:(-6, 3), (-4, 5), (-1, -2), (2, -1) and (2, 3)2.2 Find Slope and Rate of ChangeExample: Find the slope of the line passing through the points (-4,12) and (3, -2)2.3 Graph Equations of Lines: (KEY: Remember the 3 forms (slope intercept, point slope and standard form). Use the slope and y-intercept to graph. Don’t forget special examples.Example: Graph 3 + y = -2x Graph y = 5 – x Graph x = 42.4 Write Equations of Lines: (Key: These are great problems. Find the slope. Put the slope and a point into point slope form. Covert to Slope Intercept Form.Example: Write an equation of a line that passes through (-2, 5) and (-4, -1)2.5 Model Direct Variation: (KEY: Direct variation is y = ax. There is no y-intercept (b))Example: See page 1432.6 Draw Scatter Plots and Best Fitting Lines (KEY: Draw a line of best fit. Think about correlation in terms of strength (how tight the points are from the line of best fit) and direction (positive or negative). Remember that you calculator can give you the equation of the line of best fit (and r2 value) if you enter in the data to L1 and L2.Example: See page 1432.7 Use Absolute Value Functions and Transformations KEY: Remember that the absolute value function takes on a “V” shape. Shift the graph up/down, left/right, stretch/compress, and flip where needed. Example: Graph y = 3|x – 1| - 4. Compare the graph to the graph of y =|x|2.8 Graph Linear Inequalities in Two Variables (KEY: graph these just like a normal linear function but then remember that the solution is a region. You must shade above or below the line and make the line dotted or solid.)Example: Graph 3x – y ≤ -2CHAPTER 3: LINEAR SYSTEMS AND MATRICES3.1 Solve Linear Systems By Graphing: (KEY: Graph both lines and find the POINT where they intersect- Remember that your calculator can also find the intersection point after graphing both.)Example: Graph the system and estimate the solution. Check the solution algebraically.3x + y = 34x + 3y = -13.2 Solve Linear Systems Algebraically: (KEY: Use the Elimination Method and Substitution Method to Solve.)Example: Solve the system of equations algebraically (substitution or elimination).2x + 5y = 84x + 3y = -123.3 Graph Systems of Linear Inequalities: (KEY: Graph both inequalities. The solution is the REGION where the shading overlaps.)Example: Graph the system of linear inequalities3x - y ≤ 4x + y > 13.4 Solve Systems in 3 Variables: (Key: Use elimination to create 2 new equations (#4 and #5) with only 2 variables. Then you have a system of equations with 2 variables that you can solve like section 3.2. Once you have 2 variables solved for, plug these answers back into an original equation to solve for the last variable.Example: Solve the system2x + y + 3z = 5-x + 3y – 2z = 113x – y – 2z = 113.5 Perform Basic Matrix Operations (KEY: Add and Subtract Matrices of the same size by adding the terms in the same position in the matrices. Also includes multiplying by a scalar.Example: See Page 2243.6 Perform Matrix Multiplication (KEY: Row multiplied by Column! – Check if you can multiply the matrices in the first place! i.e. if your multiplying a 2x3 Matrix by a 3 x 4 Matrix. It will work (2x3)(3x4) because the middle numbers are the same. Your answer will be a 2x4 matrix (the outside numbers).Example: See Page 2243.7 Evaluate Determinants: We skipped section 3.7 but used the property in section 3.8 to use inverse matrices to solve a system.3.8 Use Inverse Matrices to Solve Linear Systems (KEY: Create a matrix of the coefficients. A matrix of the variables and a matrix of the solutions. Take the inverse of the coefficients (A-1) and multiply it by the Matrix B (the answers).Example: See Page 224CHAPTER 4: QUADRATIC FUNCTIONSStandard Form: y = ax2 + bx + cVertex Form: y = a(x – h)2 + kIntercept Form: y = a(x – p)(x – q)4.1 Graph Quadratic Functions in Standard Form: (KEY: -b/2a Gets the x-coordinate of the vertex, plug that in to get the y-coordinate). Find some other points too (like the y-intercept!)Example: Graph y = -x2 – 4x – 5 (be sure to have the vertex and y-intercept on there!)4.2 Graph Quadratic Functions in Vertex Form and Intercept Form : (KEY: see my table that I made to remember how to graph Example 1: Graph y = (x – 4)(x+2) Example 2: Graph y = -2(x + 8)2 - 34.3 Solve x2 + bx + c by FACTORING : (KEY: to factor when there isn’t a coefficient, find the factors of the “c-term” that add up to the b-term.Example: Solve x2 – 13x – 48 = 0. HINT HINT: FACTOR!!!4.4 Solve ax2 + bx + c by FACTORING : (KEY: to factor when there is a coefficient, First multiply a and c. Find the factors of that that add up to the b-term. Factor BY GROUPINGNote: This is a section where our classes could use some extra practice! Example: Solve -30x2 + 9x + 12 = 0 Hint: first divide by 3 to make this simpler.4.5 Solve Quadratic Equations by Taking Square Roots: (KEY: Isolate the x2 or a function like (x – 3)2. Then take the square root. Don’t forget that you get 2 solutions (2 real or 2 imaginary) unless you are square rooting zero (than you get 1 solution).Example: Solve 4(x – 7)2 = 80 4.6 Perform Operations with Complex Numbers : (KEY: i = -1 i2=-1)Example: Write (6 – 4i)(1 – 3i) as a complex number in standard form. 4.7 Complete the Square : (KEY: Move the c-term to the right. Add b22 to both sides. You have just made a perfect square. Write the left side as a binomial squared. Then square root. Note: if you have a leading coefficient, you must divide all terms by the coefficient as your FIRST step! Example: Solve x2 – 8x + 13 = 0 by completing the square. Example with a coefficient: 3x2 -12x + 1 = 04.8 The Quadratic Formula : (KEY: List you’re a, b, and c terms. Then plug into the quad formula and solve. Remember that you can have 2 real solutions, 1 real solution or 2 imaginary solutions).Example: Solve 3x2 + 6x = -2 using the quadratic formula.4.9 Quadratic Inequalities: (KEY: For a system of quadratic inequalities. Graph each quadratic normally. Remember to make the line dotted or solid and shade inside or outside (test a point to decide where to shade). The solution is the overlapping shaded region.Example: Graph: -2x2 + 2x + 5 ≤ 2 4.10 Write Quadratic Functions and Models: (KEY: Decide which form that is easiest to write the equation in given the provided information (i.e. are you given the vertex or the x-intercepts? Then plug in the known info to solve for “a” and write the final equationExample 1: See your book (page 322) for the picture of the graphExample 2: Write a quadratic function whose graph has the following characteristics Vertex (2, 7) and passes through (4, 2). CHAPTER 5: Polynomial FUNCTIONS (Just 5.1-5.4)We have been covering these recently so I will mainly just give you the page numbers to check out.Page 402-404 for the Review. 5.1 Properties of Exponents KEY: Use the laws of exponents to simplifyExample: (x-2y5)25.2 Evaluate and Graph Polynomials KEY: Determine the end behavior from the leading coefficient and if the degree is even or odd. Also know synthetic substitution.Example: Graph f(x) = x3 -2x2 + 3 (make a table)5.3 Add subtract and multiply Polynomials KEY: combine like terms to add/subtract. Box Method to multiply. (page 403 for more examples)Example: Perform the operation (x – 4)(2x2 – 7x + 5)5.4 Factor and Solve Polynomial Equations KEY: Factor out a GCF if possible, factor by grouping, look for special patterns.Example: Factor the polynomial completely. (page 404)x3 + 125x3 + 5x2 – 9x – 453x6 +12x4 -96x2 ................
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