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Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Algebra 1 Unit 7: Post Keystone Exam Topics Lesson 1 (PH Text 4.2, 5.8, 9.1, 7.6, 10.5): Families of Functions Lesson 2 (PH Text 9.3): Solving Quadratic Equations Lesson 3 (PH Text 9.4): Solving Quadratic Equations by Factoring Lesson 4 (PH Text 5.8): Solving Absolute Value Equations Lesson 5 (PH Text 2.5): Solving Literal Equations Lesson 6 (PH Text 10.4): Solving Radical Equations Lesson 7 (PH Text p.605): Midpoint and Distance Formulas Lesson 8 (6.0 only) (PH Text 10.2-3): Operations with radicals involving variables Lesson 9 (6.0 only) (PH Text 10.2-3): Division of Radicals involving conjugates 1 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 1 (PH Text 4.2, 5.8, 9.1, 7.6, 10.5): Families of Functions Objective: to identify families of functions for equations and graphs to predict what the graph of an equation will look like Explore - Using a Graphing Calculator: 1) Graph each function. Sketch a picture of each graph. y x2 6 y x2 y x 4 y 7x y x2 y 2x y 4x 1 y 6 x y x 3 y x2 1 y 3x 2 y 3 x y4 3 2x y 1 x2 2 y x5 2 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Summarize: 2) Sort the graphs into 5 Categories by grouping how they look. 3) What similarities among the graphs do you see in each category? 4) What differences do you see? These five Families of Functions are: Type Linear Quadratic Exponential Graph Equation Special Characteristics 3 Absolute Value Radical Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Families of Functions: Linear function highest power of x is _________ graph forms a __________________ positive coefficient of x slants _________ negative coefficient of x slants _________ Absolute Value function has an absolute value symbol around a variable expression graph forms a _________ positive coefficient of x opens _________ negative coefficient of x opens _________ Quadratic function highest power of x is _________ graph forms a __________________, or a _________ positive coefficient of x opens _________ negative coefficient of x opens _________ Exponential function In the form y = a · bx, where a ≠ 0, b > 0, b ≠ 1, and x is a real number graph forms a __________________ positive coefficient of x opens _________ negative coefficient of x opens _________ Radical function highest power of x is _________ graph forms a _____________________________ positive coefficient of x opens _________ negative coefficient of x opens _________ 4 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Class Practice: Sketch an example of each of the following: 1.) Quadratic 2) Linear 3) Radical Identify the Function Family to which each belongs. Describe how you knew: 4) y 5 x 4 5) y 7 45 x 6) f ( x) 5x 7 7) y = 6x2 + 1 8) y = 4x – 1 9) y = x2 + 3x + 2 10) y = 3x 11) y 8x 12) y = 7 – x 5 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 2 (PH Text 9.3): Solving Quadratic Equations Objectives: to solve quadratic equations by graphing and using square roots A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0, where a ≠ 0. The standard form of a quadratic equation is ax2 + bx + c = 0. Quadratic equations can be solved in a variety of ways. We will consider graphing, using square roots, and factoring. Solving quadratic equations by graphing: Graph the function. The x-values of points where the graph crosses the x-axis are considered the solutions. Solving quadratic equations by using square root: Isolate the squared term. Find the square roots of each side, and simplify. 2x2 – 98 = 0 HW: p.551 #20-36 even 6 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 3 (PH Text 9.4): Solving Quadratic Equations by Factoring Objectives: to solve quadratic equations by graphing and using square roots Finding solutions to quadratic equations is also an important use of factoring. A replacement set is the set of all solutions to a polynomial equation. Any real numbers that make an equation a true statement are a part of the equation’s replacement set. The easiest way to find an equation’s solution set is to apply the zero-product property. Zero-Product Property: For all real numbers a and b, if ab = 0 then a = 0, b = 0 or both a and b = 0. To find the replacement set to a given equation: 1. Write the equation in standard form. 2. Factor the polynomial completely. 3. Set each of the factors equal to zero. 4. Solve for the variable. 5. Check by replacing the potential solution for the variable in the equation. If it results in a true statement, it is a part of the equation’s replacement set. Example: 8x2 + 10x – 3 factored is (2x + 3) (4x – 1) so (2x + 3) = 0 Check: Replacement set for 8x2 + 10x – 3: Find the possible solutions for 4x2 – 21x = 18 HW: p.558 #8-36 even 7 and/or (4x – 1) = 0 { } Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 8 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 9 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 4 (PH Text 3.7): Solving Absolute Value Equations Objective: to solve equations that involve absolute value Some equations may have two solutions. One time this can happen is when the variable is within absolute value bars. two solutions one solution no solution x5 x 5 Examples: x 5 x = 5 or -5 x=5 absolute value can never be negative Class Practice: x 16 1) 2) x 5 11 5) 6 3) 3 w 4 6) x 5 9 4) 4 n 32 7) x9 3 9) x 2 8) p3 4 0 Jeff estimates his stride is 16 inches. However, any given stride is likely to vary from this estimate by up to 2 inches. Write and solve an equation to find Jeff’s minimum and maximum stride length. HW: p.211 #10-30 even 10 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 5 (PH Text 2.5): Solving Literal Equations Objective: to use the strategies of reciprocals and opposites to solve equations involving variables only A literal equation is an equation that expresses a relationship among variables. Formula - shows the relationship between two or more variables You can transform (change) a formula to define a different variable by “solving for” that variable. Use the skills we learned to solve equations. Example: I = prt I p rt Class Practice: Solve for the underlined variable. Show your work! 1) x 10 y 2) 3(2a b) c 3) h 2(l 2m) 5) a 2b 4b 3c 6) x y 5 x 2 Solve each for the given variable. 7) P 2l 2w for w 8) 1 A bh 2 11 for b 9) d = rt for t Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 12 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 6 (PH Text 10.4): Solving Radical Equations Objective: to solve equations containing radicals; to identify extraneous solutions. For any real number n, n2 n Remember: Simplify. a) 90 b) 4 18 c) 3( 6 2 8) b) 27x3 c) 50a6 Example: Simplify. a) 72 y 2 Class Practice: Simplify. 1) 8b8 2) 63x3 3) 5c5 4) 300x6 13 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Example 2: Evaluate 2n 10 a) for n = 3 b) for n= 9 Class Practice: Evaluate for the given variable. Then simplify, if possible. 5) c 7; c 15 Example 3: For what values of y will 6) 5 x 6; x 6 2 y 5 be a real number? Class Practice: Find all values of x that make each radical expression a real number. 7) 5x 10 8) 9) 4 x 2 10) 14 2x 7 3x 9 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics My Class Practice: Find all values of x that make each radical expression a real number. 1) 2) x 3 11 4x 7 1 3) 6x 4 4x 6 4) x 4 x 12 5) 12 6 x 6) 3x 13 7 x 3 7) 2 x 3 13 8) 3x 8 x 6 15 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 16 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 7 (PH Text p.605): Midpoint and Distance Formulas Objectives: To find the distance between two points in a coordinate plane; To find the midpoint of two points. 1. Graph the points ( –3, 4 ), ( 1, 1 ), ( –3, 1) and connect them to form a triangle. Mark the lengths of the legs by counting units. Use the Pythagorean Theorem to find the length of the hypotenuse. a 2 b2 c2 Now use the distance formula to find the length between (1,1) and ( –3, 4 ). d= x2 x1 2 y 2 y1 2 The distance formula: For points P x1 , y1 and Q x2 , y2 in the coordinate plane, the distance d between the points is given by: d= x2 x1 2 y 2 y1 2 Round answers to the nearest tenth!!!! 2. Find the distance between (1, 4) and (−2, −5). 3. Find the distance between (−3, 2) and (3, −2). 4. One hiker is 4 miles west and 3 miles north of the campground. Another is 6 miles east and 3 mile south of the campground. How far apart are the hikers? (the camp ground is at (0, 0) ) 17 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 5. Mickey travels 15 miles west, then 20 miles north. Jamie travels 5 miles east, then 10 miles south. How far apart are they? 6. Quadrilateral KLMN has vertices with coordinates K(-3, -2), L(-5, 6), M(2, 6) and N(4, -2). a. Show that LK MN . b. Use slopes to show that LK and MN are parallel. The midpoint of a segment is the halfway point between two endpoints. The coordinates of a midpoint are the averages of the coordinates of the endpoints. The midpoint formula: For endpoints P x1 , y1 and Q x2 , y2 on the coordinate plane the midpoint m can be expressed by: x x y y M = 1 2 , 1 2 2 2 8. Find the midpoint of P(−1, 6) and Q(5, 0) 7. Find the midpoint of A(2, -1) and B(4, -3) HW: p.605 #1-6 18 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics x2 x1 y2 y1 The distance formula: d The midpoint formula: x x y y2 M 1 2, 1 2 2 2 19 2 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 8 (6.0 only) (PH Text 10.2-3): Operations with radicals involving variables Objective: to add, subtract, multiply and divide radicals involving variables. To add and subtract, simplify first, then combine like terms. Example: Simplify. Make sure all answers are in simplest radical form. 1) 100 x 2 y 144 x 2 y 5 x 2 y 2) 18 x 2 y 8 x 2 y 50 x 2 y Remember when multiplying, multiply the coefficients (number outside) then multiply the radicands (the number inside). Simplify your answers if possible. Example: Simplify. Make sure all answers are in simplest radical form. a a 3) 7 y 3 8 y 4) 5a b b To divide, simplify first using power rules, then rationalize the denominator if necessary. Example: Simplify. Make sure all answers are in simplest radical form. 5) 1 x7 6) 20 24 x 2 y 3 xy 2 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Class Practice: Perform the indicated operations. Express your answers in simplest radical form. 2) 175a2b 3 112a2b 72 xy 2 2 98 xy 2 1) 3) 3 81ab2 2 ab2 5 9ab2 4) 3 n 5) 2 8xy 3xy 6) 8) 7) 9) x 2 y 2 27 x 5 3x 10) 21 2 a5b bc2 3 a3 cd c3d 3 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 22 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 23 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 24 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Lesson 9 (6.0 only) (PH Text 10.2-3): Division of Radicals involving conjugates Objective: Be able to rationalize denominators using conjugates. The radical expression 3 2 2 is called the conjugate of 3 2 2 . You use this to rationalize the denominator of a fraction when adding and subtracting is involved. If m and n are non-negative, then the binomials a m b 2 and a m b 2 are conjugates. The sum and difference of the same two terms. Class Practice: Write the conjugate of each binomial. 1) 1 2 3) 5 3 5 2 ) 2 3 Example: Rationalize the denominator. 5 a) 3 2 Class Practice: Rationalize the denominator and simplify. 2 5 1 4) 5) 6 3 5 3 7) 5 2 11 2 8) 7 3 10 5 25 b) 2 5 3 3 2 2 6) 2 3 1 5 9) 3 22 3 2 5 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Section 4: Solving Radical Equations. Objective: Be able to solve radical equations A radical equation is an equation that contains radicals with variables in the radicand. To solve: 1) 2) 3) 4) Isolate the radical on one side of the equation and combine any like terms Square both sides to eliminate the radical Repeat steps 1 and 2 if necessary. Check your answer. Example: Solve and Check. 1) 3 a 1 5 2) 3) 3 3x 2 5 x 26 5t 2 16 t Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics Class Practice: Solve and check. 1) a4 6 3) 6 3 y 0 2) 4x 4 0 4) 5x 1 3 7 5) 3x 4 1 4 6) 3n2 12 3n 7) 2m 5 7 3 8) 5 y2 7 2 y 27 Algebra 1 Mrs. Bondi Unit 7 Notes: Post Keystone Exam Topics 28