Advanced Algebra Chapter 3 - Note Taking Guidelines

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1 Advanced Algebra Chapter 3 - Note Taking Guidelines 3.1 Constant-Increase or Constant-Decrease Situations 1. What type of function can always be used to model a Constant-Increase or Constant-Decrease Situations 2. Study example 1 3. Now try the following problem: a. Claire sells sports cars. She gets a base salary of $30,000 a year plus 2% of her sales. If Claire s sales for the year total D dollars, what will her salary be for the year? 4. In an equation of the form y = mx + b: a. Which variable is the dependent variable? b. Which variable is the Independent variable? c. What is the generic ordered pair for this equation? d. Is y a function of x? 5. What are the two names we give to the point at which a linear equation crosses the y-axis? 6. What is the domain and range of a slanted (oblique) line?

2 7. What is the rate of change of a linear model called? 8. What is the form y = mx + b called? 9. What is a linear function? 10. What can be used to describe every constant-increasing or constant-decreasing Situation? 11. Study example Now try the following problem: a. Rick gets an allowance of $15 per week. Whenever his parents pick up a dirty dish he has left behind, Rick loses $0.30. i. Write an equation modeling this situation. ii. Graph the equation from part i. iii. If Rick received no allowance last week, how many dirty dishes did his parents pick up?

3 13. What two things do we know about the graph of a constantincreasing situation? 14. What two things do we know about the graph of a constantdecreasing situation? 15. When the rate of change for a linear situation is zero, what type of line do we have? 16. What two things can you tell about a linear function when it is written in the form y = mx + b? 17. The slope of a linear equation can also be referred to with what phrase?

4 18. What is a piece-wise linear function? 19. Study example Now try the following problem: a. The graph below describes Leah s weight over the first 16 weeks of her life. Write a story explaining the meaning of each segment of the piecewiselinear graph. Summarize what you learned in sections 3.1:

5 3.2 The Graph of y = mx + b 1. What are the three ways you should be able to graph a line in the form y = mx + b? 2. Study example 1 3. Now try the following problem: a. Graph each equation using its slope and y- intercept: i. y = 2x 3 2 y = x + 3 ii What does the slope of a line tell us about the line? 5. What are the steps for graphing a line written in y = mx + b form? 6. In order to write a linear equation in slope-intercept form we need to 7. Study example 2 8. Now try the following problem: a. Graph the line 3y = 2x - 15

6 9. What do we know about lines with negative slope? 10. If the slope of a line is positive, what do we know about the line? 11. What kind of line has zero slope? 12. Study example Now try the following problem: a. Graph the line y = -3 b. What is the domain and range of the above function? 14. Write the slope-intercept equation, y = mx + b, for a horizontal line. 15. What do we know about two lines with the same slope? 16. What do we know about two non-vertical lines that are parallel? Summarize what you learned in sections 3.2:

7 3.3 Linear-Combination Situations 1. Describe what we know about the variables in a linear combination. 2. Study example 1 3. Now try the following problem: a. Becky sold $36 worth of tickets for the key club picnic. Adult tickets cost $5 and student tickets cost $2. There is no charge for children under 6 years of age. i. Write an equation to express the relationship among the number of adult tickets, student tickets and children's tickets Becky sold. ii. Make a table and graph all possibilities that could have occurred.

8 4. Study example 2 5. Now try the following problem: a. Mixtures A and B contain weed killer and water. Mixture A is 5% weed killer, and mixture B is 15% weed killer. i. Write an equation relating A, B, and T, the total amount of the weed killer. ii. How many ounces of 5% solution must be added to 50 ounces of 15% solution to get 12 ounces of weed killer in the final solution? iii. Draw a graph to illustrate all possible (A, B) that give 12 ounces of weed killer in the final mixture. Continuous Domain - Discrete Domain - Summarize what you learned in sections 3.3:

9 3.4 The Graph of Ax + By = C 1. What is the Standard Form for the equation of a linear function? 2. What do we know about the values of A, B, and C in the Standard Form? 3. What is the graph of the equation Ax + By = C when A and B are not both zero? 4. The slope of a line in Standard Form Ax + By = C is equal to what fraction? 5. How can we find the y-intercept of a line when it is written in standard form? 6. What kind of line do we end up with if A = 0? 7. Give the equation and type of line when B = Study example 1 9. Now try the following problem: a. Graph the line x + 0y = -3 b. Find the slope of the line.

10 10. What is the slope of any vertical line? 11. Summarize the three types of lines. 12. Which lines are graphs of functions? 13. When graphing an equation in standard form it may be easier to find the x-intercept and y-intercept. a. How do we find the y-intercept? b. How do we find the x-intercept? 14. What is a y-intercept? 15. What is an x-intercept? 16. Study example Now try the following problem: a. Graph the equation 10x + 6y = 30 by using its intercepts. Summarize what you learned in sections 3.4:

11 3.5 Finding an Equation of a Line 1. How many points determine a line? 2. Study example 1 3. Now try the following problem: a. Suppose you know that the formula relating blood pressure and age is linear and that normal systolic blood pressures are 110 for a 20-year-old and 130 for a 60-year-old. Construct a formula in which blood pressure B is a function of age A. 4. What is the benefit of having a linear function written in the Slope-y- Intercept form(y = mx + b)? 5. Identify the Point-Slope form for the equation of a linear function? 6. What is the benefit of having a linear function written in the Point- Slope form(y y1) = m(x x1)? 7. Describe the given information you would need in order to decide on using the Slope-y-Intercept form.

12 8. Describe the given information you would need in order to decide on using the Point-Slope form. 9. Study example Now try the following problem: a. Find an equation for the line L through (-3, 6) and (5, 0). 11. Given two points of a line how do we go about finding the equation for the linear function? 12. Study example Now try the following problem: a. The cost of running the Bayshore Hotel is $ 2250 per day when 25 rooms are occupied and $5250 when 125 rooms are occupied. i. If the relationship between the number of occupied rooms r and the cost c of running the hotel is linear, write the equation relating c and r. ii. Find the cost of running the hotel when 75 rooms are occupied.

13 14. Given the slope and a point of a line how do we go about finding the equation for the linear function? 15. Given the slope and the y-intercept of a line how do we go about finding the equation for the linear function? 16. How do we find the equations for a Piecewise linear function. 17. Study example Now try the following problem: a. According to a chart of the Blue Star Life Insurance Company, the lightest recommended weight for an adult woman with a medium frame and height of 4' 10" is 109 lb. This weight increases 2 lb./in to height of 5' 1", then it goes up to 3 lb./in to a height of 6'. i. Draw a graph showing how the weight in pounds w for an adult woman with a medium frame is related to her height in inches h. ii. Find two equations that together describe the relation between the lightest recommended weight w in pounds for a given height h in inches. Summarize what you learned in sections 3.5:

15 3.6 Fitting a Line to Data 1. If everyone in class tried to fit a line to a set of data points by hand would we all come up with the same equation? 2. What is a scatterplot? 3. Study example 1 4. Now try the following problem: a. Use the line through the points A = (1900, 1500) and B = (2000, 2500) to fit a line to the data points for the population of Kansas (in thousands) for each decade year from the 1900 census through the 1990 census. 5. What is the result of fitting a line to data? 6. Study example 2 7. Now try the following problem: a. Predict the population of Kansas in the year 2025 using: i. The equation from Now Try #4 above. ii. The equation for the regression line.

16 8. How is the correlation coefficient used when working with lines of best fit or linear regression? 9. What would the correlation coefficient be when fitting a line to data using only two data points? 10. Study example Now try the following problem: a. Use linear regression to find an equation of the line through the points (0, 32) and (100, 212). Summarize what you learned in sections 3.6:

17 3.7 Recursive Formulas for Arithmetic Sequences 1. What is an arithmetic sequence? 2. In an arithmetic sequence what is true about the difference between any term and the preceding term? 3. Define the recursive definition for an arithmetic sequence. 4. Study example 1 5. Now try the following problem: a. Consider the sequence generated by: a1 = 53 a n = a n 1 7 for integers n 2. i. Describe this sequence in words. ii. Write the first five terms of the sequences.

18 6. Study example 2 7. Now try the following problem: a. Due to an increasing population, the town of Valley Heights is concerned about its water supply. The town council has voted to immediately add 20,000 acre-feet of water to its reservoir capacity of 3 million acre-feet and to add an additional 20,000 acre-feet of water each year. Write a recursive formula to express the capacity of the reservoir in n years. 8. When working with an arithmetic sequence the first term is referred to with what phrase? 9. An arithmetic sequence represents a constant increasing or constant decreasing situation. a. What is the other name given to an arithmetic sequence? b. What does this tell us about all the points generated by the sequence? Summarize what you learned in sections 3.7:

19 3.8 Explicit Formulas for Arithmetic Sequences 1. When is an explicit formula for a sequence better than a recursive formula? 2. Define the explicit formula for an arithmetic sequence. 3. Study example 1 4. Now try the following problem: a. Find an explicit formula for the arithmetic sequence: 12, 14.5, 17, 19.5 b. Find a Study example 2 6. Now try the following problem: a. Find the 75 th term of the arithmetic sequence: 27, 28.5, 30, Study example 3 8. Now try the following problem: a. The first row in an auditorium has 15 seats in it. Each subsequent row has 3 more seats than the row in front of it. If the last row has 78 seats, how many rows are in the auditorium? Summarize what you learned in sections 3.8:

21 3.9 Step Functions 1. What does the graph of a step function look like? 2. How do you know a step function is a function? 3. Definition of Greatest Integer Function: 4. Study example 1 5. Now try the following problem: ( ), evaluate each of the following a. If g x = 2 + x functions: i. g(2.6) ii. g(-3.1) iii. g (π )

22 6. What is the notation for the greatest integer function? 7. What is another name for the greatest integer function? 8. Study example 2 9. Now try the following problem: ( = x a. Graph f x) Study example Now try the following problem: a. Banks often put pennies in rolls of 50. How many full rolls can be made from p pennies? Summarize what you learned in sections 3.9:

Section 1.1: Functions and Models

Section 1.1: Functions and Models Definition: A function is a rule that assigns to each element of one set (called the domain) exactly one element of a second set (called the range). A function can be