1.2 Explain Solving Multi-Step Equations - Notes Essential Question: How can you use multi-step equations to solve real-life problems? Main Ideas/ Questions Notes/Examples What You Will Learn To solve multi-step linear equations using inverse operations. To use multi-step linear equations to solve real-life problems. Solving Equations using Inverses: Addition/Subtraction Multiplication/Division **Always do the same thing to both sides of the equation, but only once on each side** What you already know Solution: A solution makes the equation true. Practice: Decide if the following values are solutions to the given equations by substituting the value in for the variable. 1. 3x = 12, x = 4 YES/NO 2. x 3 = 10, x = 7 YES/NO Steps to Solve Equations Step 1: Use the distributive property to remove any grouping symbols. Step 2: Simplify the expression on each side of the equation. Step 3: Collect the variables on one side of the equation and the constant terms on the other side. Step 4: Isolate the variable (add/subtract then multiply/divide). Practice: Solve each equation using inverses. Show all work. 3. 3x + 4 = 19 4. Check your 5. 4z 10.4 = 2.4 6. Check your solution to #3. solution to #5.
1.2 Explain Solving Multi-Step Equations - Notes 7. 17 = z ( 9) 8. f 4 5 = 9 9. q+( 5) 3 = 8 10. 5x + 3x = 28 11. 28 = 13.8x + 262.6 12. 5 1 (x 6) = 4 2 13. 4(z 12) = 42 14. 33 = 12r 3(9 r) 15. 7 + 3(2g 6) = 29 Application: 16. Your school s drama club charges $4 per person for admission to a play. The club borrowed $400 to pay for costumes and props. After paying back the loan, the club has a profit of $100. a. Write an equation. Let x represent the number of people who attended the play. b. How many people attended the play?
1.3 Explain - Solving Equations with Variables on Both Sides - Notes Essential Question: How can you solve an equation that has variables on both sides? Main Ideas/ Questions What You Will Learn Notes/Examples Solve linear equations that have variables on both sides Identify special solutions of linear equations Use linear equations to solve real-life problems The Ultimate Goal Goal: to get the variable you are solving for on one side of the equation by itself with a coefficient of positive one. Example: x = 5 Solving Equations with Variables on Both Sides Steps to Solve Equations Step 1: Use the distributive property to remove any grouping symbols. Step 2: Simplify the expression on each side of the equation. Step 3: Collect the variables on one side of the equation and the constant terms on the other side. Step 4: Isolate the variable (add/subtract then multiply/divide). Practice: Solve each equation using inverses. Show all work. 1. 12 3x = 6x 2. Check your 3. 7 5z = 17 + 5z 4. Check your solution solution to #1. solution to #3 5. 8 + 6x 10x = 16 8x 6. 9x 12 = 1 (32x + 56) 4
1.3 Explain - Solving Equations with Variables on Both Sides - Notes 7. 2.4(2x + 3) = 0.1(2x + 3) 8. 9 2(5d + 4) = 3(2d 7) 10 Special Solutions If you lose your variable, the solution is special Application: Equations may have.. One solution - (EX. x = 3) Infinitely Many Solutions identity an equation that is true for all values of the variable (Ex. 2 = 2) No Solution an equation that is not true for any value of the variable (Ex. 6 = 0) Practice: Solve. 9. 5x + 2(5x + 3) = 15x 10. 2(4y + 1) = 8y 2 The statement is never true. So, the equation has solution. The statement is always true. So, the equation is an and has solutions. 11. You and your friend drive toward each other. The equation 50h = 5(38 9h) represents the number h of hours until you and your friend meet. When will you meet?
2.2 and 2.3 Explain Solving Inequalities - Notes Main Ideas/ Questions Essential Question: How do you solve an inequality? Notes/Examples What You Will Learn To solve inequalities. To use inequalities to solve real-life problems An inequality is.. Inequalities A solution of an inequality is List 2 solutions of the inequalities: 1. x > 4 x = x = Solving Inequalities 2. y < 3 y = y = Steps to Solve Inequalities Step 1: Use the distributive property to remove any grouping symbols. Step 2: Simplify the expression on each side of the equation. Step 3: Collect the variables on one side of the equation and the constant terms on the other side. Step 4: Isolate the variable (add/subtract then multiply/divide). ***When solving inequalities, you must reverse the inequality sign when multiplying or dividing by a negative*** Practice: Solve the inequality. k 3. 6 9 u 2 4. 3(x + 2) 12 5. 2 2
2.2 and 2.3 Explain Solving Inequalities - Notes 6. 14 7y 7. 4b 1 > 7 8. 8 th Grade Review: Graph the solution to #7 on a number line. Write the sentence as an inequality. Then solve the inequality. 9. Six is less than or equal to the sum of a number and 15. 10. Six is more than the difference of a number d and 1. Application 11. You bike for 2 hours at a speed no faster than 17.6 miles per hour. a. Write and solve an inequality that represents the possible numbers of miles you bike. b. The bike portion of an Ironman competition is 112 miles. Your friend says that if you continue to bike at this pace, you will be able to complete the bike portion of the Ironman in less than 6.5 hours. Is your friend correct? Explain.
2.4 Explain Solving Multi-Step Inequalities - Notes Main Ideas/ Questions Essential Question: How can you solve a multi-step inequality? Notes/Examples What You Will Learn To solve multi-step inequalities. To use multi-step inequalities to solve real-life problems. Steps to Solve Inequalities Step 1: Use the distributive property to remove any grouping symbols. Step 2: Simplify the expression on each side of the equation. Step 3: Collect the variables on one side of the equation and the constant terms on the other side. Step 4: Isolate the variable (add/subtract then multiply/divide). ***When solving inequalities, you must reverse the inequality sign when multiplying or dividing by a negative*** Practice: Solve each inequality. 1. 2 + b 3 2. 8 1.2(d + 1) 3. 5 2n > 8 4n 3 4. 8j 4j + 6 < 2 + 3j 5. 12( 1 w + 3) 2(w 4) 6. Graph the solution in #1. 4 Solve: 5 2n > 8 2n Does every equation have a solution? Making Connections
2.4 Explain Solving Multi-Step Inequalities - Notes Inequalities with Special Solutions Practice: Solve the following inequalities. 6. 8b 3 > 4(2b + 3) 7. 2(5w 1) 7 + 10w If you lose your variable, the solution is special The inequality is. So, there is. The inequality is. So, are solutions. 7. You need to withdraw cash from the bank for an upcoming concert. a. Write an inequality that represents how many $20 bills you can withdraw from the account without going below the minimum balance. b. Is it reasonable for you to be able to withdraw 12 $20 bills without going below the minimum balance? Justify your answer. Solving Real- Life Problems 8. You need a mean score of at least 90 points to advance to the next round of the touch-screen trivia game. What scores in the fifth game will allow you to advance?
1.4 Explain - Rewriting Equations and Formulas - Notes Essential Question: How can you use a formula for one measurement to write a formula for a different measurement? Main Idea/Questions What You Will Learn Notes/Examples Rewrite literal equations Rewrite and use formulas for area Rewrite and use other common formulas Definition: an equation that has two or more variables Practice: Determine if the following are literal equations. Explain your answer. Vocabulary 1. A = 1 bh 2. 3x + 14 = 21 3. 3x 2y = 12 2 Rewriting Literal Equations Practice: Solve the following literal equations for the indicated variable. 4. 3y + 4x = 9, solve for y 5. 20x + 5y = 15, solve for y 6. 6x 3y = 6, solve for y 7. P = 2l + 2w, solve for w
1.4 Explain - Rewriting Equations and Formulas - Notes 8. A = 1 bh, solve for h 9. C = 5 (F 32), solve for F 2 9 10. I = Prt, solve for P 11. d = rt, solve for r 12. The formula for a pitcher s earned run average, or ERA, is a = 9r p run average, r is earned runs, and p is innings pitched. where a is the earned a) Solve the formula for r. b) Find the earned runs for a pitcher with an ERA of 2.63 who has pitched 89 innings. Question: How does solving for a specific variable in a literal equation compare to solving an equation for the specific value of an unknown?
3.1 and 3.2 - Day 1- Explain Functions - Notes Essential Question: What is a function? How do you find the domain and range of discrete data? Main Ideas/ Questions What You Will Learn Notes/Examples To determine whether relations are functions. To find the domain and range of a function in mathematical problems and realworld situations. 8 th Grade Review: What is a function? Each input has exactly one output - CANNOT repeat!! Vertical Line Test Practice: Determine if each relation is a function. 1. YES/NO 2. YES/NO 3. YES/NO Function Not a Function Types of Data Discrete Data (dots) Data that involves a count of items, such as the number of people or the number of cars. On a graph indicated with a point (dot).on a graph the points are disconnected. Continuous Data Data where numbers between any two data values have meaning, such as measurement of temperature, length, or weight. On a graph indicated with a solid line or segment, on a graph the points are connected. Practice: Determine if the given is a function and if it is discrete or continuous. 4. Function? YES / NO 5. Function? YES / NO 6. Function? YES / NO DISCRETE/CONTINUOUS DISCRETE/CONTINUOUS DISCRETE/CONTINUOUS The battery power remaining on a smartphone at any given time. (# of Suitcases, Total Weight) 7. Function? YES / NO 8. Function? YES / NO 9. Function? YES / NO DISCRETE/CONTINUOUS DISCRETE/CONTINUOUS DISCRETE/CONTINUOUS (Time, Temperature Outside) 10.Function? YES / NO DISCRETE/CONTINUOUS y = 2 11.Function? YES / NO DISCRETE/ The depth of a scuba diver CONTINUOUS returning to the surface of an ocean after a certain amount of time.
3.1 and 3.2 - Day 1- Explain Functions - Notes Domain and Range for Discrete Data How are domain and range written for discrete data? Mathematical Domain and Range The set of all possible values of the independent and dependent variables. 12. {(-1, 9), (5, 9), (2, 7), (4, -2)} 13. The DOMAIN is x: { } The RANGE is y: { } The DOMAIN is x: { } The RANGE is y: { } 14. DOMAIN: { } 15. DOMAIN: { } RANGE: { } RANGE: { } 16. The function y = 3x + 12 represents the amount y (in fluid ounces) of juice remaining in a bottle after you take x gulps. a. The domain is 0, 1, 2, 3, and 4. What is the range? Reasonable Domain and Range The set of all possible values of the independent and dependent variables that make sense in a real-world situation. 17. John s gift card is loaded with $12. He plans to use his gift card to buy books that cost $3 each. (, ) Equation: Reasonable Domain: Why is this function a discrete function? Reasonable Range: 18. Graduation tickets are $15.95 per person. Each student can buy a maximum of three tickets. (, ) Equation: Reasonable Domain: Why is this function a discrete function? Reasonable Range: 19. Joe has an afterschool job at the local sporting goods store. He makes $6.50 an hour. He always works at least 1 hour but never more than 5 hours in a week. Joe must work the full hour to get paid. (, ) Equation: Reasonable Domain: Why is this function a discrete function? Reasonable Range:
3.1 and 3.2 Explain Functions Day 2- Notes Essential Question: How do you find the domain and range of continuous data? Main Ideas/ Questions What You Will Learn Notes/Examples To find the domain and range of a function in mathematical problems and realworld situations. Domain and Range for Continuous Data How is it written for continuous data? Open Circle Closed Circle Practice: Find the Domain: Look at the x-axis. 1. x Practice: Find the Range: Look at the y-axis. 4. y 5. y 6. y Domain: 2. x Domain: 3. x Domain: Range: Range: Range: Write #1 using words.. Write #6 using words. Practice: Find the domain and range. 7. 8. 9. 10. Domain: Domain: Domain: Domain: Range: Range: Range: Range:
3.1 and 3.2 Explain Functions Day 2- Notes Reasonable Domain and Range The set of all possible values of the independent and dependent variables that make sense in a real-world situation. 11. Jane has 3 cereal bars in her backpack. A cereal bar contains 130 calories. The number of calories consumed is a function of the number of bars eaten. a. Define Variables: (, ) b. Write an equation to represent the total number of calories consumed: c. Reasonable Domain: d. Reasonable Range: Discrete and Continuous Functions Practice: Determine if the given functions are discrete or continuous and then find the domain and range. 12. 13. 14. 15. Discrete/Continuous Discrete/ Continuous Discrete/ Continuous Discrete/Continuous Domain: Domain: Domain: Domain: Range: Range: Range: Range: 16. Billiard World charges $5 to rent a pool table plus $10 per hour of game play. Customers are charged for the full hour. The total amount charged to rent the table is a function of the number of hours the table is rented. John plans on renting the pool table for at least two hours but no more than 4. a. Discrete or Continuous? Explain. b. Define Variables: (, ) c. Write an equation to represent the total cost to rent a table: d. Reasonable Domain: e. Reasonable Range: 17. A fundraising organization will donate $250 plus half of the money it raises from a charity event. a. Discrete or Continuous? Explain. b. Define Variables: (, ) c. Write an equation to represent the total amount that will be donated: d. Reasonable Domain: e. Reasonable Range:
3.3 Explain - Function Notation - Notes Essential Question: How can you use function notation to represent a function? Main Ideas/ Questions What You Will Learn Notes/Examples To use function notation to evaluate and interpret functions To use function notation to solve and graph functions To solve real-life problems using function notation What is function notation? Old way y = mx + b (x, y) in the equation is replaced with. New way f(x) = mx + b (x, f(x)) f(x) represents the output and is read as: the value of f at x or f of x Letters other than f can be used to name a function, such as g(x) or h(t) Practice: Write the following equations in function notation. 1. y = 5x + 3 2. y = 3x 100 3. y = 2x 2 + 3x 4 Using Function Notation to Evaluate and Interpret Practice: Evaluate the following function given the value, and then write an order pair to represent the function. 4. Evaluate f(x) = 4x + 7 when f(2) and f( 2) f(2) = (, ) f( 2) = (, ) 5. Evaluate f(x) = x 2 + 2x + 1 given f( 3) f( 3) = (, ) Let f(t) be the outside temperature ( F) t hours after 6 A.M. What do we know? (, ) (, ) Explain the meaning of each statement. 6. f(0) = 58 7. f(6) = n 8. f(3) < f(9)
3.3 Explain - Function Notation - Notes Evaluating for the Independent Variable 9. h(x) = 2x 7 when h(x) = 2 When h(x) = 2, x =. (, ) 10. g(x) = 2 x 5 when g(x) = 7 When g(x) = 7, x =. (, ) 3 Evaluating Functions Using Graphs, Tables, and Real World Applications Practice: Given the following graph, 11. Find f(2) - (This means find f(x) when x = 2) Put your finger on the graph where x = 2. What is the y value for that point? 12. f( 2) = So, f(2) = 13. Now work backwards. If f(x) = 4, what is x? 1 2 14. a) Complete the table using the function f(x) = 2x + 5. b) f( 3) = c) f(0) = d) If f(x) = 1, find x. x -5-3 -2-1 0 f(x) 15. The function f(x) = 350 125x represents the number of miles a helicopter is from its destination after x hours. a. How far is a helicopter from its destination after 1.5 hours? b. How many hours has the helicopter been in the air at 100 miles?
3.5 and 4.1 Explain Graphing and Writing in Slope-Intercept Form - Notes Essential Question: How can you describe the graph of the equation y = mx + b? Given the graph of a linear function, how can you write an equation of the line? Main Ideas/ Questions What You Will Learn Notes/Examples To interpret slopes and y-intercepts of linear equations. To use slopes and y-intercepts to solve real-life problems To write equations in slope-intercept form given table of values, graphs, and verbal descriptions. x and y represent the ordered pairs that are contained on the line. Slope is. m = rise change in y = = y run change in x x Also called the Rate of Change.. change in the dependent variable change in the independent variable slope with units, Ex.: miles, cost gallons item Top unit depends on the bottom unit (ex.: miles depend on gallons), feet second The y-intercept is.. sometimes called the constant, the starting amount, or the initial amount. b = x y 1 1.9 0 1-1 0.2 1. The graph shows the height y (in meters) of a helicopter as it descends in x seconds. Find and interpret the rate of change and y-intercept of the given line. Rate of Change: y-intercept: Represents: Represents: Identifying and Interpreting Slopes From Tables Slope of a Line: y 2 y 1 x 2 x 1 for 2 points (x 1, y 1 ) and (x 2, y 2 ) Practice: Determine the slope for each of the following. 2. 3. 4. The table shows the number of songs purchased, s, and the total cost C (in dollars). Find and interpret the rate of change of the line. Rate of Change: Represents:
3.5 and 4.1 Explain Graphing and Writing in Slope-Intercept Form - Notes Graphing Linear Equations Practice: Find the slope and the y-intercept of the graph of the linear equation and then graph. 5. 4x + 2y = 12 6. 3x + y = 0 7. 2x + y = 2 m = b = m = b = m = b = 8. A linear function f models a relationship in which the dependent variable decreases 6 units for every 3 units the independent variable decreases and f(0) = 4. a. Graph this function b. Slope: c. y-intercept: Writing Linear Equations Practice: Write an equation in slope-intercept form for each of the following linear functions. 9. 10. 11. 12. Write the equation of a line in slope-intercept form that passes through (-3, 5) and (0, -1). 13. An electrician charges $120 after 2 hours of work and $190 after 4 hours of work. a. What is the electrician s initial fee? b. How much does the electrician charge per hour? c. Write a linear model that represents d. What is the total cost for 7 hours of work? the total cost as a function of the number of hours worked.