Math 165 Section 3.1 Linear Functions

Transcription

1 Math 165 Section 3.1 Linear Functions - complete this page Read the book or the power point presentations for this section. Complete all questions on this page Also complete all questions on page 6 1) Write the equation of the line 2) Is the average rate of change (ARC) of a linear function constant or variable? 3) Geometrically, what does the ARC of the linear function represent? Write the symbolic representation shown in the power point. 4) Read the two examples shown on the power point (or in the book): one is about population and the other about maximum number of heart beats. Which of them is a linear function? How do you recognize if a table displays a linear pattern? 5) How do you recognize if a linear function is increasing, decreasing or constant? Summarize the three cases and give an example of each. Show graph. 1

2 Math 165 Section 3.1 continued Reading Graphs Linear Functions Solving linear equations and inequalities graphically 6) Book-problems, page # 32 - Use the graphs to solve the equations and inequalities 7) Book-problem, page # 36 - Use the graphs to solve the equations and inequalities 2

3 Section 3.2 Finding Linear Models for Linearly Related Data Reviewing the linear regression process learned in your algebra class the steps are outlined below You will apply this technique when solving the problem shown on the next page. The steps to get the linear regression line are outlined below Here are the steps for Linear Regression There is also a video in my website that shows you how to do this. GET HELP if needed To see the data plot (Scatter-diagram or scatter-plot): Linear Regression - Calculator Instructions 1. Hit STAT 2. Then choose EDIT and press enter a. To clear a list, go up to the name of the list, press CLEAR, ENTER 3. Put the independent variable (x-values) data in L1 (2 nd 1) 4. Put the dependent variable data (y-values) in L2 (2 nd 2) 5. Hit 2 nd Y= (STAT PLOT) 6. Choose Plot1 7. Turn it on 8. Choose the first icon under Type 9. Make sure Xlist says L1 10. Make sure Ylist says L2 11. It doesn t matter which mark you choose 12. Hit ZOOM 9 (ZOOMSTAT) 13. This will show you the data plot To get the equation (the line of best fit): 1. Hit STAT 2. Choose CALC 3. Choose #4 LinReg 4. Hit 2 nd 1 (L1) 5. Hit comma (for older calculators only) 6. Hit 2 nd 2 (L2) 7. Hit comma (for older calculators only) 8. Hit VARS 9. Choose Y-VARS 10. Choose Function 11. Choose Y1 12. Hit ENTER For Older calculators it looks like LinReg(ax+b) L1,L2,Y1 For newer calculators it looks like this: You now have the equation on your home screen and in the Y= under Y1. 3

4 Math Section 3.2 Finding Linear Models for Linearly Related Data 8) #20, page 146, our book 4

5 Section 3.2 Finding Linear Models for Linearly Related Data 9) #22, page 147, our book 5

6 Section 3.3 Quadratic Functions complete this page Read the book or the power point presentations for this section. Complete all questions on this page 10) Write the general form of a quadratic function. a. What is the domain? b. What is the name of the graph? c. If the quadratic coefficient is positive, then d. If the quadratic coefficient is negative, then 11) How do you find the x-coordinate of the vertex? 12) How do you find the y-coordinate of the vertex? 13) Given that the coordinates of the vertex are V(h, k) write the vertex form of a quadratic function. 14) Discuss the possibilities of the range of a quadratic function. Show graphs. 15) To find the x-intercepts of a quadratic function we use the formula: 16) What are the possibilities of the number of x-intercepts of a quadratic function? Show graphs. What will be the discriminant for each case? 6

7 Math 165 Section 3.3 Applications of Quadratic Functions 17) Throwing an Object on the Earth/Moon The height of an object which is falling or is projected into the air on the Earth's surface is given by 2 h(t) = -16t o o + vt+ s, where h is the height of the object (in feet), o per second), s o is the original height (in feet), and t is the time (in seconds). v is the original velocity of the object (in feet The position equation on the moon is given by + vt+ s 2 h(t)=-2.7t o o PART 1 - MOON - An astronaut standing on the surface of the moon throws a rock into the air at an initial velocity of 30 ft/sec. Assuming his hand is 6 feet from the surface of the moon, a) Write the quadratic function. b) Sketch the graph, no calculator. Label axes in context. c) When does the rock reach its maximum height? d) What is the maximum height of the rock? e) How long will the rock remain in the air? f) When will the rock be higher than 70 feet? Solve graphically. (section 3.5) 7

8 Math 165 Section 3.3 Applications of Quadratic Functions 18) Problem from last page continued PART 2 - EARTH Assume the rock is thrown on the surface or the Earth with an initial velocity of 30 ft/sec. and the hand of the person is 6 feet from the surface of the Earth. a) Write the quadratic function. b) Sketch the graph, no calculator. Label axes in context. c) When does the rock reach its maximum height? d) What is the maximum height of the rock? e) How long will the rock remain in the air? f) When will the rock be higher than 10 feet? Solve graphically. (section 3.5) 8

9 Math 165 Section 3.4 Building Quadratic Models Enclosing the Most Area with a Fence 19) A farmer with 100 meters of fencing wants to enclose a rectangular plot that borders a river; he will not fence the side along the river. Let x be the side perpendicular to the river. a) Sketch the plot and label. Use L for the side parallel to the river b) Analyze the area of some rectangular plots that border the river according to the given specifications. Side perpendicular to river, x Side parallel to river Area Length of fence that has been used c) Plot the points (x, A) in a coordinate system, Label axis in context. b) What type of function models this data? c) According to the table, what is the value of x for which the area is maximum? d) What is the maximum area? e) What is the length of the side parallel to the river? 9

10 Math 165 Section 3.4 Building Quadratic Models 20) Enclosing the Most Area with a Fence - Let s do a similar problem analytically. A farmer with 3000 meters of fencing wants to enclose a rectangular plot that borders a river; he will not fence the side along the river. Let x be the side perpendicular to the river. a) Sketch the plot, label, use L for the side parallel to the river. b) Using the variables shown on you sketch, write the formula for the area of the rectangular plot. c) Write a function that gives the area of the rectangular plot in terms of x. d) Analytically, find the value of x that optimizes the area. e) Find the largest area. f) Find the other dimension of the rectangular field. g) Sketch the graph of the area function. Label axes in context. label the coordinates of the vertex. 10

11 Math 165 Section 3.4 Building Quadratic Models 21) Enclosing the Most Area with a Fence - A farmer with 2,000 meters of fencing wants to enclose a rectangular plot and needs to fence the four sides. a) Write an equation for the area of the rectangular plot as a function of the width of the rectangle, x. b) Use your knowledge of quadratic functions to sketch the graph of the function. Label axes in context. c) Use algebra to find the value of x that maximizes the area. d) What is the maximum area? e) What are the dimensions of the plot of largest area? f) Use your calculator to graph and check your answers. 11

12 Math 165 Section 3.4 Building Quadratic Models 22) Enclosing the Most Area with a Fence - A farmer with 3,000 yards of fencing wants to enclose a rectangular plot and needs to fence the four sides and include also two interior fences parallel to the width x. g) Write an equation for the area of the rectangular plot as a function of the width of the rectangle, x. h) Use your knowledge of quadratic functions to sketch the graph of the function. Label axes in context. i) Use algebra to find the value of x that maximizes the area. j) What is the maximum area? k) What are the dimensions of the plot of largest area? l) Use your calculator to graph and check your answers. 12

13 Math 165 Section 3.4 Building Quadratic Models 23) #6, page 166, our book a) Express the revenue R as a function of price, p. b) What price will maximize the revenue? c) What is the maximum revenue? d) How many units should be sold to maximize revenue? e) Sketch the revenue function, label axes in context. f) At 8 dollars per unit, what will the revenue be? What will be the number of units sold? 13

14 Math 165 Section 3.4 Building Quadratic Models 24) #6, page 166, our book 14

15 Math 165 Section 3.4 Building Quadratic Models 25) Constructing Rain Gutters - A rain gutter is to be made of aluminum sheets that are 18 inches wide by turning up the edges 90 degrees. What depth x, will provide maximum cross sectional area and hence allow the most water to flow? a) Produce an equation for the area of the cross sectional region as a function of x. b) Sketch the function and label in context. c) Algebraically determine the dimensions that will produce the maximum cross sectional area. d) What is the maximum cross sectional area? e) Now, use a graph to check your answers. Label axes in context. 15

16 Math 165 Section 3.5 Inequalities with Quadratic Functions, Graphical Approach 26) Solve graphically 16

17 Math 165 Section 3.5 Inequalities with Quadratic Functions; Analytical Approach 27) Solve analytically 17

18 Math 165 Section 3.5 Inequalities with Quadratic Functions; Applications 28) Solve the following problems 18