Increasing and Decreasing Functions oh my Standard MCC
- Slides: 44
Increasing and Decreasing Functions, oh my! Standard: MCC 8. F. 5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e. g. , where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Essential Questions: How can we describe different linear functions? How can we determine what the graph will look like based on information given?
Rollercoaster Ride! http: //www. youtube. com/watch? v=0 Wp. NSImh 6 Z 8 With a partner, discuss the following… If we were to graph the route of the rollercoaster, how would you describe the graph?
The Ups and Downs • Think of a function as a roller coaster going from left to right • Uphill – Slope > 0 – Increasing function • Downhill – Slope < 0 – Decreasing function 4
Definitions • Given function f defined on an interval – For any two numbers x 1 and x 2 on the interval • Increasing function – y 1 < y 2 when x 1 < x 2 • Decreasing function X 1 X 2 f(x) – y 1> y 2 when x 1< x 2 5
Linear or Nonlinear? A B C
Linear or Nonlinear? Increasing or Decreasing?
Linear or Nonlinear? Increasing or Decreasing?
Linear or Nonlinear? Increasing or Decreasing?
Which of the following is true about the data in the table? Celsius: 0 5 10 15 Fahrenheit: 32 41 50 59 A. The data is nonlinear and decreasing B. The data is nonlinear and increasing C. The data is linear and decreasing D. The data is linear and increasing
Bivariate Data • Bivariate means “two variables”, in other words there are two types of data. • Examples: What are the two variables? 1. An ice cream shop keeps track of how much ice cream they sell versus the temperature on that day. 2. Ms. Janas wonders if her students’ test scores are impacted by the amount of time her students spend playing video games.
Quantitative Data vs. Qualitative Data • Deals with descriptions. • Data can be observed but not measured. • Colors, textures, smells, tastes, appearance, beauty, etc. Qualitative data: • blue/green color, gold frame • smells old and musty • texture shows brush strokes of oil paint • peaceful scene of the country • masterful brush strokes Quantitative Data • Deals with numbers. • Data which can be measured. • Length, height, area, volume, weight, speed, time, temperature, humidity, sound levels, cost, members, ages, etc. Quantitative data: • picture is 10" by 14" • with frame 14" by 18" • weighs 8. 5 pounds • surface area of painting is 140 sq. in. • cost $300
Example Qualitative data: • robust aroma • frothy appearance • strong taste • burgundy cup Quantitative data: • 12 ounces of latte • serving temperature 150º F. • serving cup 7 inches in height • cost $4. 95
Example Qualitative data: • friendly • civic minded • focused • positive school spirit Quantitative data: • 672 students • 394 girls, 278 boys • 68% on honor roll • 150 students accelerated in mathematics
Describe yourself… Quantitative Data Qualitative Data
Bivariate Frequency (Two-way) Tables • Bivariate frequency tables compare two variables x y # of shakes # of candies left 0 35 1 26 2 14 3 7 4 4 5 0
Investigating Scatter Plots What type of correlation do you think these graphs have?
Scatter Plots Scatter plots are most often used to show correlations or relationships between two sets of data. X-axis – independent variable Y-axis – dependent variable
Positive Correlation • Positive correlations occur when two variables or values move in the same direction. • As x increases , y increases x↑ y ↑ • As x decreases, y decreases x↓ y↓
Ex: As the number of hours that you study increases your overall class grade increases. Study Time Class Grade 0 55 0. 5 61 1 67 1. 5 73 2 81 2. 5 89 3 91 3. 5 93 4 95 4. 5 97
Negative Correlation • Negative Correlations occur when variables move in opposite directions. • As x increases , y decreases x↑ y↓ • As x decreases, y increases x↓ y↑
Ex: As the number of days per month that you exercise increases your actual weight decreases. Work out time Weight 0 200 0. 5 205 1 190 1. 5 195 2 180 2. 5 190 3 170 3. 5 177 4 160 4. 5 170 5 150
No Correlation • No correlation exists if there is no noticeable pattern in the data
Ex: There is no relationship between the number of shirts someone owns and their annual salary number of shirts owned salary 1 1 2 0 3 50 4 30 5 25 6 17 7 2 8 40 9 8 10 25 11 12 12 7 13 19 14 55 15 71 16 9
Outliers • An outlier is an observation or value that lies outside of the overall pattern of a set of data.
Clustering • Clustering occurs when data is gathered around a particular value.
Positive, Negative, or No Correlation?
Question 1 Positive correlation means that as x increases, y ______ or as x decreases, y ______.
Question 2 Read the following scenario and determine if the data is bivariate. If so, state the two variables being compared. Ms. Jones has 35 students in her class and she uses roughly 4 reams of paper week. Ms. Rogers has a smaller class of 10 students and uses half a ream of paper week. Ms. Hall has 25 students and uses 2 reams of paper week.
Question 3 Create a scatter plot with the following data, tell what type of correlation the data has. DO NOT ERASE YOUR GRAPH, you will need it for the next question! # of students Reams of paper used per week 35 4 15 1 20 1. 5 10 . 5 35 3 25 2 30 10
Question 4 Using your graph from question 3, identify the outlier(s) if any.
Question 5 Negative correlation means that as x increases, y ______ or as x decreases, y ______.
Question 6 What type of correlation does this graph have? How do you know? (Remember math vocabulary)
Question 7 Sort the following into two categories: quantitative variables and qualitative variables. 4 wheels 22” rims 5 passengers red leather interior weighs 1175 kilograms safe 35 mpg made by Ford
Question 8 Give one example of data that could be quantitative or qualitative, depending on how you describe it. Ex. She is wearing a red shirt. (Qualitative) The red shirt that she is wearing costs $20. (Quantitative)
Line of Best Fit • A line on a graph showing the general direction that a group of points seem to be heading. • A line of best fit may also be called a trend line since it shows us the trend of the data • The line may pass through some of the points, none of the points, or all of the points.
Example: Temperature vs. Ice Cream Sales Are the variables quantitative or qualitative? Identify each variable as dependent or independent.
How do we find the line of best fit? 1. Step 1: Plot data on the graph. 2. Step 2: Draw a line through the data that you feel best represents the pattern of data. 3. Step 3: Choose two coordinates on the line. 4. Step 4: Find the slope using the two coordinates that you chose. 5. Step 5: Find the y-intercept. 6. Step 6: Write the equation for the line.
Example •
Example •
Sometimes, the two coordinate points that the line of best fit go through are not data points. Is the above equation the correct equation for this line of best fit?
Standards and EQs Standards MCC 8. SP. 2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line MCC 8. SP. 3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. EQs How can we analyze, interpret, and compare data?
Making Predictions using Line of Best Fit If the line of best fit for the following data is y = 30 x + 100, how we use this equation to predict ice cream sales when the temperature is 30ºC?
Example x = number of hours spent studying y = grade on a test Find the line of best fit that goes through the points (3, 85) and (5, 95). Predict your grade on the test if you study 2 hours.
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