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Unit 4 Lesson 1: Linear vs. Non-Linear Functions1st What’s a function?From graph – passes the vertical line testFrom a table of points – cannot have 2 x’s the sameFrom a mapping diagram – for every output there is exactly 1 input-889003022600 Function Not a Function3359150571500How do recognize a linear function from an equation, graph or table of points?Equation: Slope Intercept Form y = mx + b Standard Form ax + by = cGraph:Table of Points:(The FIRST difference in the y-values is constant). Example #1: Justify the table of points represents a linear function.How do you recognize a quadratic function from an equation, graph or a table of points?Equation:Standard Form: ax2 + bx + cVertex Form: a(x – h)2 + kGraph:Table of Points:(The SECOND difference in the y-value is constant).Example #2: Justify the table of points represents a quadratic function.How do you recognize an exponential function an equation, graph or table of points?Equation: y = abxGraph:Table of points:(The ratio of y-values is constant).Example #3: Justify the table of points represents a exponential function.Example #4: Decide whether the table of points represent a linear, quadratic, exponential function or neither.You Try….Decide whether the table of points represent a linear, quadratic, exponential function or neither.Unit 4 Lesson 2: Slope & Linear Functions DefinitionExample(s)Independent VariableThe variable whose change is NOT affected by the another variableAge; TimeDependent VariableThe variable that is being measuredHeight; weightSlopeMeasures the steepness of a lineriserun= y2- y1x2- x1From a graphFrom an equationy = mx + bRate of ChangeThe constant change in the y-values. The constant change could result in the constant difference of the y-values (Linear), the SECOND constant difference in the y-values (Quadratic), or the constant ratio in the y-values (Exponential).Linear:Quadratic:Exponential:y-interceptThe point where the point crosses the vertical axis.From a graph:From an equation:Y=mx + bUnit RateThe real-world interpretation of slope.Examples: Identify the slope and y-intercept from an equation. Then, graph. Y = -3x + 28x – 2y = -14You Try…Y = 23x – 3 4x + 3y = -24Example: Identify slope & y-intercept from a table of points. Then Graph.Example: Identify the slope & y-intercept from a graph.Special LinesHorizontal LinesSlope is zero riserun = 0runY = a constant valueExample: y = 5 or 4y = -12Vertical LinesSlope is undefined riserun = rise0X = a constant valueExample: x = -3 or 4x = 1VocabularyExample #1: Independent vs. dependent variable.#1:#2You Try….Example #2: Unit Rate & Slope.At what rate is the diameter increasing?Based on the information, what is the price of the soda?Who walks faster, Carlos or Lucy?Who walks faster, Laura or Piyush?You Try……Example #3: Interpret the linear real-world word problem below:Ralphie makes $8 per hour. Complete the table of points.Number of hours (x)012456Wages (y)Graph the function.What’s the constant change?How much will Ralphie make if he works 36 hours?Unit 4 Lesson 3: Graphing & Linear FunctionsVocabularyDefinitionExample(s)DomainThe set of all possible x-valuesDomain: Discontinuous54485746190 Domain: ContinuousD= -6 ≤x <8RangeThe set of all possible y-valuesRange: Discontinuous5407993953600Range: ContinuousR = -7 ≤x ≤4x-interceptPoint on the x-axisExamples: Interpreting Linear Word ProblemsThe function below shows the cost of a hamburger with different numbers of toppings.C = 1.90 + 1.40tState the independent variable.State the dependent variable.State the domain. What does it mean?State the range. What does it mean?State the y-intercept. What does it mean?If Jodi paid $3.30 for a hamburger, how many toppings were on the hamburger?Megan and her family are traveling from their home in Nashville, TN to Orlando, FL on a Disney vacation. The trip is 685 miles and they will be traveling 65 miles per hour, on average. Megan used the following equation to calculate the remaining distance throughout the trip.D = 685 – 65hState the x-intercept, y-intercept and slope. Give the meaning of each in the context of the problem.________________________________________________________________________________________________________________________________________________________________________________________________________________________State the independent and dependent variables.________________________________________________________________________________________________________________________________________________State the domain and range. Give the meaning of each in the context of the problem.________________________________________________________________________________________________________________________________________________You try….Unit 4 Lesson 4: Writing Linear EquationsExample #1: Writing a linear equation given the slope and y-intercept. (y = mx + b)M = 5 and y-intercept = -6Example #2: Writing an equation given 1 coordinate and slope.m = 5 at (-4, 3)Example #3: Writing an equation given 2 coordinates.(2, -5) (-3, 4)You Try… Write the Equation(-3, 1) (2, 0)M = 5 and (8, -12)Slope = -2 and y-intercept = 4(5, 6) (4, 3)Unit 4 Lesson 5: Writing Linear Equations – Real WorldExample #1: Writing a linear equation given the slope and y-intercept. (y = mx + b)A video rental store charges a $20 rental fee and $2.25 for each video rented. Identify the unit rate.What is the cost if you do NOT rent a video?Write the equation.If 15 videos were rented, what’s the revenue?If a new member paid $67.50 how many videos were rented?Example #2: Writing an equation given 1 coordinate and slope.Marty is spending money at the average rate of $3 per day. After 14 days he has $68 left. The amount left depends on the number of days that have passed. Write an equation for the situation.Find the amount of money he began with.How much money does Marty have after 9 days?Example #3: Writing an equation given 2 coordinates.Suppose a 5-minute overseas call costs $5.91 and a 10-minute call costs $10.86. The cost of the call and the length of the call are related. The cost of each minute is constant.What is the cost, c, of a call of m minutes duration? How long can you talk on the phone if you have $12 to spend?You Try….Unit 4 Lesson 6: Linear RegressionDefinitionExample(s)Linear RegressionA line that has an equation of the form Y = b + aX, where X is the explanatory variable and Y is the dependent variable.Y = 2.23 + 4.2xScatterplotA?graph?of?plotted?points that show the relationship between two sets of data. In this example, each dot represents one person's weight versus their height.Scatter?(XY)?Plots.Line of Best fit?A?line of best fit?(or "trend"?line) is a straight?line?that?best represents the data on a scatter plot. This?line?may pass through some of the points, none of the points, or all of the points.?Example #1: The scatter plot shows the costs y of bottles containing x fluid ounces of juice.How much does a gallon of juice cost?4412394141909How many fluid ounces of juice can you purchase for $3?Draw a line that you think best approximates the points. Write an equation for your line.Use the equation to predict the cost of a 256-fluid ounce container of juice.Does the data show a positive, a negative, or no relationship?Example #2: 49775173843100The scatter plot shows the relationship between the numbers of girls and the numbers of boys in 10 different classrooms.What type of relationship, if any, does the data show?Is it possible to find the line of fit for the data? Explain.Is it reasonable to use this scatter plot to predict the numberof boys in the classroom based on the number of girls? Explain.Example #3The table shows the numbers of losses y a gamer has x weeks after getting a new video game.3919993647150029781565405Week, x1234567Losses, y151210763100Week, x1234567Losses, y1512107631Make a scatter plot of the data.Draw a line of fit.Write an equation of the line of fit.Does the data show a positive, a negative, or no relationship?Interpret the relationship.You Try….The scatter plot shows the weights y of an infant from birth through x months.right889000At what age did the infant weigh 11 pounds?What was the infant’s weight at birth?Draw a line that you think best approximates the points. Write an equation for your line.Use the equation to predict the weight of the infant at 18 months.Does the data show a positive, a negative, or no relatio ................
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