unit 2 Functions

Standard Topic

7 Distinguish between functions & relations 8 Use Function & evaluate for inputs in their domain.

9a Write the arithmetic sequence with given information 9b Use arithmetic sequence to model a situation

10a Write equations in two variables to model a situation 10b Write equations of a line from a graph 10c Write equations of a line from a table or coordinates 10d Write equations of a line given two points 11 Graph the equation of a line 12 Choose the best line of fit

13a Interpret slope in context 13b Interpret intercepts in context 14a Calculate linear regression model & correlation 14b Interpret the correlation coefficient 15 Distinguish between correlation & causation

2.1 Introduction of and evaluating functions.

Paste the relation in the box to indicate function or not function.

Function Not a function

HSF.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). HSF.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. HSF.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

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Function – a relationship where each input (x) has ONLY ONE matching output (y)

Is a value of x repeated?

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raph

M Run an imaginary vertical line across you graph. M If the vertical line EVER

touches the graph in more the one place, the graph is NOT A FUNCTION.

M If the vertical line

ALWAYS touches the graph in ONLY on location, the graph is a FUNCTION.

Evaluate Functions Function Notation f(x) = 2x-1

example: g(x)= x2 – 5 find g(-1)

h(x) = -3x+2 find h(10)

vertical l ine test

2.1 Introduction of and evaluating functions Practice Determine if each relation is a function.

1. 2. 3.

4. 5. 6. 7. {[6, -3), (7, 4), (-7, -2), (0, -2)} 8. {(7, 1), (7, -3), (7, 4)} 9. 10. 11. Evaluate the function. 12. If f(x)=2x-3, find f(-2) 13. If f(x)=2x-3, find f(7) 14. If k(x)=-7x+1, find k(0) 15. If k(x)=-7x+1, find k(-1) 16. f(x)=2x-3 and f(x)=15, find the value of x.

x y -3 1 3 1 0 2 2 0

x y -6 36 -3 9 0 0 3 9

x y 1 -4 2 4 1 -4 -2 3

Review Problems (complete ALL) 17. Solve for y: 5x-12y=24 18. Solve for x: 4x – 3(x+2)= -5x +2 19. solve for x: !

!!≥ 5 20. 𝑥 + 4 = 12

21. Solve for x: 6x-2(x-2)=-3x+5 22. solve for y: 2x-3y=15 23. solve for x: 𝑥 − 5 = 10

2.2 Arithmetic Sequence

11, 8, 5, 2, … What would the 21st term be?

Example: 20, 24, 28, 32, 36, …

{-38, -45, -52, -59, …} Find the 52nd term

Find the explicit formula for {-18, -9, 0, 9, …}. Find the recursive formula for {26, 24, 22, 20, …}.

an = a1 + (n-1) d

HSF.BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

example

recursive Using a previous term to calculate the next term.

explicit A formula that allows

you to compute any term of a sequence.

an=an-1+d

an=a1+(n-1)d

Practice

Arithmetic Sequence Word Problems After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day?

A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the 20th row of the theater?

Brian gets a starting wages of $15 and a annual raise of $1.50 per hour. What will Brian’s hourly wage be during his tenth year? (Hint: How many years has he worked when he starts out earning $15?)

Starting May 1, a new store will begin giving away 500 posters as a promotion. Each day, 4 posters will be given away. If the store is open 7 days a week, how many posters will the store have left when it opens for business on May 14? Days of promotion 1 2 3 4

Posters remaining 500 496 492 488

2.2 Arithmetic Sequence Practice 1. Find the 10th term of each of the following arithmetic sequences.

{19, 25, 31, 37 … } {101, 97, 93, 89…} 2. Find the 15th term of each of the following arithmetic sequences.

{31, 36, 41, 46…} {5, -3, -11, -19…}

3. Consider the sequence {87, 83, 79, 75…}

a. Show that the sequence is arithmetic (does it have a common difference? What is the common difference?)

b. Find the explicit (general) form.

c. Find the recursive form.

d. Find the 40th term e. Which term of the sequence is -297?

4. Consider the sequence {6, 17, 28, 39, 50…}

a. Show that the sequence is arithmetic.

b. Find the explicit (general) form.

c. Find the recursive form.

d. Find the 40th term

e. Which term of the sequence is 330? 5. Edgar is getting better at math. On his first quiz he scored 57 points, then he

scores 61 and 65 on his next two quizzes. If his scores continued to increase at the same rate, what will be his score on his 9th quiz? Show all work.

a. Write an explicit formula for the sequence. Explain where you found the numbers you are putting in the formula.

b. Identify the value of n and explain where you found it. Use the explicit

formula to solve the problem.

c. Write your final answer as a sentence.

6. Sherry works at the local fast food chain at the rate of $32.50/hr. The management said that, depending on her performance, her hourly wage will be increased by $7.50/hr every month. In how many months will she be receiving $85 an hour?

7. There is a stack of logs in the backyard. There are 15 logs in the 1st layer, 14 in the

second, 13 in the third, 12 in the fourth, and so on with the last layer having one log. How many logs are in the stack?

8. Jerry deposited $20,000 on an investment that will give $1,750 for every year that

his money stays in the account. How much money will he have in his account by the end of year 8?

Review Problems (complete ALL) 9. Evaluate the function f(x)=5(x+3)-2 given f(3). 10. Solve for y: 3x+4y=-16 11. Solve for m: !

!= 2

12. 𝑥 + 3 = −5 13. 2 𝑥 +−4 = 10 14. 4-3(2x-5)=12x-3 15. -12x + 10 ≥ -10x+2 16. Determine if the function is a relation 17. Determine if the function is a relation then explain. then explain.

2.3 Write, graph, & interpret equations (and word problems) given slope & y-intercept.

y= m x +b

Slope-Intercept Form

∆𝑦∆𝑥 =

𝑟𝑖𝑠𝑒𝑟𝑢𝑛 =

(𝑦! − 𝑦!)(𝑥! − 𝑥!)

Independent variable

Substitute x to find the

y-value.

y-intercept

Where the line crosses

the y-axis

example y=2x+1 y=-!

!x +5

HSA.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSF.LE.A.2 Construct linear and exponential functions, including arithmetic given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). HSF.IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

HSF.LE.B.5 Interpret the parameters in a linear function in terms of a context.

Slope-intercept practice Write the equation of the line that passes through (2, 1) with a slope of 3. Verify that the equation passes through the given points by graphing.

Write the equation of the line that passes through each pair of points (3, 1) & (2, 3) Verify that the equation passes through the given points by graphing.

Write the equation of the line that passes through each pair of points (5, -8) & (-7, 0) Verify that the equation passes through the given points by graphing.

Write the equation of the line that passes through (4, -7) with a slope of -1. Verify that the equation passes through the given points by graphing.

The table shows the number of domestic flights in the U.S from 2004 to 2008. Write an equation that could be used to predict the number of flights if it

continues to decrease at the same rate.

Ten people from a local youth group went to Black Hills Whitewater Rafting Tour Company for a one-day rafting trip. The group paid $425. Write an equation to find the total cost C for p people.

How much would it cost for 15 people?

2.3 Write, graph, & interpret equations (and word problems) given slope & y-intercept. Practice Graph each equation 1. 𝑦 = − !

!𝑥 + 2 2. 3𝑦 = 2𝑥 − 6 3. 6𝑥 + 3𝑦 = 6

4. Carla has already written 10 pages of a novel. She plans to write 15 additional pages per month until she is finished. Write an equation to find the total number of pages P written after any number of months, m. Graph the equation on the grid. Find the total number of pages written after 5 months. Write an equation of the line that passes through the point and has the given slope. 5. (-4, 1) slope -1. 6. (2, 6) slope 2 7. (5, -3) slope !

!

Write an equation of the line that passes through each pair of points. 8. (9, -2), (4, 3) 9. (-5, 3), (0, -7) 10. (-1, -3), (-2, 3) 11. In 1904, a dictionary cost 30 cents. Since then the cost of a dictionary has risen an average of 6 cents per year. Write a linear equation to find the cost C of a dictionary y years after 1904.

If this trend continues, what will the cost of a dictionary be in 2020? 12. Jackson is ordering tickets for a concert online. There is a processing fee for each order, and the tickets are $52 each. Jackson ordered 5 tickets and the cost was $275. Determine the processing fee. Write a linear equation to represent the total cost C for t tickets. Make a table of values for at least three other numbers of tickets. Graph this equation. Predict the cost of 8 tickets. 13. A plumber charges $25 for a service call plus $50 per hour of service. Write an equation in slope-intercept form for the cost, C, after h hours of service. What will be the total cost for 8 hours of work? 10 hours of work? 14. A caterer charges $120 to cater a party for 15 people and $200 for 25 people. Assume that the cost, y, is a linear function of the number of x people. Write an equation in slope-intercept form for this function. What does the slope represent? How much would a party for 40 people cost? 15. Which equation best represents the graph?

a. 𝑦 = 2𝑥 b. 𝑦 = −2𝑥 c. 𝑦 = !

!𝑥

d. 𝑦 = − !!𝑥

Review Problems (complete ALL) 16. Evaluate the function: f(x) = -!

!x + 5 when f(x) =-6

17. −3 𝑥 + 2 − 7 ≤ 4(−𝑥 + 4) 18. Solve for y: -4x+6y=-3 19. If ℎ 𝑥 = !

!𝑥 − 3 find ℎ(6) 20. !

!!< 12

21. Consider the sequence {4, -2, -8, -14, -20,…} a. Show that the sequence is arithmetic.

b. Find the explicit (general) form.

c. Find the recursive form.

e. Find the 29th term

e. Which term of the sequence is -98?

22. Determine if the relation is a function and explain why. 23. −2 2𝑥 + 4 = −12 24. Tori deposited $5,000 on an investment for college that will give $175 for every year that her money stays in the account. How much money will 2he have in her account by the end of year 12?

-5 6 3 -2

3 2 1

2.4 Write, graph, & interpret equations (and word problems) given two points.

Given a Slope and A Point m=3 (-3, 4)

drop in the slope and the point distribute & solve for y

(y -y1) = m (x -x1)

Given two Point (-5, 4) (3, -2)

calculate slope drop in the slope and the one of the points distribute & solve for y

(y -y1) = m (x -x1) Calculate slope

Given an Application Problem

The first month a company was open, it had 2 employees. At the end of 6 months, the company had 10 employees. If the number of employees increases at a steady rate, write an equation

that illustrates this situation.

Write the coordinates ( , ) ( , )

calculate slope fill in equation

(y -y1) = m (x -x1)

simplify equation

2.4 Write, graph, & interpret equations (and word problems) given two points.

2.4 Write, graph, & interpret equations (and word problems) given two points. Practice Write an equation in point-slope form based on the given information. Then convert the equation to slope-intercept form. 1. point: (-2, 5), slope: -6 2. point: (4, 3), slope: − !

!

3. points: (-1, 7), (8, -2) 4. points: (-4, 3), (0, 1) 5. point: (-6, -8), 𝑚 = − !

! 6. (-2, 11), 𝑚 = !

!

Write an equation in point-slope form to model each situation. Then re-write the point-slope form to slope-intercept form. 7. The number of copies of a movie rented at a video kiosk decreased at a constant rate of 5 copies per week. The 6th week after the movie was released, 4 copies were rented. How many copies were rented during the second week? 8. Write an equation in point-slope form for the line containing 𝐺𝐻. Write the slope-intercept form of the line.

9. The barometric pressure is 598 millimeters of mercury (mmHg) at an altitude of 1.8 kilometers and 577 millimeters of mercury at 2.1 kilometers. Write a formula for the barometric pressure as a function of the altitude. What is the altitude if the pressure is 567 millimeters of mercury? 10. You bought 8 pencils for $3.35 on Monday. Your friend bought 10 pencils for $4.25 on Friday. Write an equation to model the total bill y, in dollars, for the number of pencils, x. 11. At 7am I leave for school, which is 30 miles away. At 7:15am I am 18 miles away. Write an equation in point-slope form to model this situation and then find the distance I am away from the school after 25 minutes. (y is the distance away from the school and x is the time) 12. Andrew is a submarine commander. He decides to surface his submarine to periscope depth. It takes him 20 minutes to get from a depth of 400 feet to depth of 50 feet. Write an equation in point-slope form that describes this situation. What was the submarine’s depth five minutes after it started surfacing? Review Problems (complete ALL) 13. 𝑓 𝑥 = 2𝑥! − 1 find f(3) 14. Is this a function? Explain why or

why not. 15. Timmy has 3, 000 miles on his car. He drives 38 miles each day. Write a linear model to represent the situation. How many miles will he have on his car after 20 days? 16. Given the points (-1, 5) & (3, 7) (a) Find the slope (b) Write the equation of the line.

x y -2 2 0 1 -2 0 1 -1

2.5 Line of Fit and Extrapolation

Example A biologist is studying the relationship between a

tree’s diameter and its height. She records the

following data for 7 different

trees. How to enter data into lists and graph a scatterplot

Put data into your calculator

M Press and choose 1: Edit M Clear this list if it isn’t empty . M Type the data in L1& L2 M Set up a scatterplot in the Statistics Plot menu

M Go to Plot 1 and adjust the settings as shown.   ZoomStat

How to calculate & interpret linear regression Linear Regression Equation

M Press then enter. Use this information to write an equation. a=slope, b=y-intercept. y=3 . 179x+1 .536

Enter this equation into calculator and choose again.

Diameter (in.)

Height (ft.)

2 8 3 10 4 16 5 17 6 22 7 20 8 29

S-ID 6

a – Fit a function to the data; use functions fitted to data to solve problems in the

context of the data. Use given functions or choose a function suggested by the

context. Emphasize linear, quadratic, and exponential models.

b– Informally assess the fit of a function by plotting and analyzing residuals.

c– Fit a linear function for a scatter plot that suggests a linear association.

Diameter is the independent variable (x)

and Height is the dependent variable (y).

We will calculate and interpret the regression line of fit and correlation coefficient.

How to calculate & interpret correlation coefficient

Turn Diagnostic on This step only has to been ONCE unless you reset your calculator.

Run Linear Regression again. You now have r= which is the correlation.

Correlation Co – meaning together and relation

Correlation has strength (high to low) and direction

(positive or negative) Strong Correlation is good!

R=.9588 is high positive

2.5 Line of Fit and Extrapolation Practice Select the best line of fit for the data. 1. 2. 𝑦 = −𝑥 + 4.5 𝑦 = 10𝑥 + 20 𝑦 = − !

!𝑥 + 4 𝑦 = 20𝑥 + 10

For each of the following situations, calculate the regression line & correlation. Interpret the slope and y-intercept in context of the problem and describe the correlation. (You wi l l complete the starred problems after the NEXT lesson) 3. The accompanying table shows the enrollment of a preschool from 1980 through 2000. (Remember, in order to interpret y-intercept we must use 0, 5, 10, … for years instead of 1980, 1985, etc.) Linear Regression Equation: M What does slope mean in this context? M What does y-intercept mean in this context? Correlation coefficient: Describe correlation:

4. The accompanying table shows the percent of the adult population that married before age 25 in several different years. Linear Regression Equation:

M What does slope mean in this context?

M What does y-intercept mean in this context? Correlation coefficient: Describe correlation: Using the equation found above, estimate the percent of the adult population in the year 2009 that will marry before age 25, and round to the nearest tenth of a percent. 5. A factory is producing and stockpiling metal sheets to be shipped to an automobile manufacturing plant. The factory ships only when there is a minimum of 2,050 sheets in stock. The accompanying table shows the day, x, and the number of sheets in stock, f(x). Linear Regression Equation:

M What does slope mean in this context?

M What does y-intercept mean in this context? Correlation coefficient: Describe correlation: Use this equation to determine the day the sheets will be shipped.

6. Since 1990, fireworks usage nationwide has grown, as shown in the accompanying table, where t represents the number of years since 1990, and p represents the fireworks usage per year, in millions of pounds.

Linear Regression Equation:

M What does slope mean in this context?

M What does y-intercept mean in this context? Correlation coefficient: Describe correlation: Using this equation, determine in what year fireworks usage would have reached 99 million pounds. Based on this linear model, how many millions of pounds of fireworks would be used in the year 2008? Round your answer to the nearest tenth.

7. The accompanying table illustrates the number of movie theaters showing a popular film and the film's weekly gross earnings, in millions of dollars.

Linear Regression Equation:

M What does slope mean in this context?

M What does y-intercept mean in this context? Correlation coefficient: Describe correlation: Using this linear regression equation, find the approximate gross earnings, in millions of dollars, generated by 610 theaters. Round your answer to two decimal places. Find the minimum number of theaters that would generate at least 7.65 million dollars in gross earnings in one week. Review Problems (complete ALL) 8. Solve: 2(x-1) + 3x = 4x 9. Solve & graph : -2 ≤ 4x-2 ≤ 10 10. Is this a function? Explain why or why not. 11. Find the explicit formula. {(2, 3), (4, 3), (5, 5)} {-2, 3, 8, 13}

12. Write the equation of the line with a slope of -2 that passes through (1, -3).

13. Graph 3x-2y=10

2.6 Correlation or Causation & Interpreting Slope & Intercepts From the biologists tree study from lesson 2.5 Regression Line: y=3.179x + 1.536

!.!"#  !"  !"  !"#$!%!  !".!"#$%&%'

words: For every inch in diameter the tree grows, it’s height will increase, on average, 3,179 ft.

b-1.536 words: When the diameter of the tree is 0 in, the height would be 1.536 feet. substitute y=0 words: When the height of the tree is 0 in, solve for x the diameter would be -.483 feet. x=-.483

The following graph depicts the relationship between the sales of

ice cream and the temperature according to the weather

recorded each day.

Mexican Lemon imports prevent highway deaths.

Is there any relationship between student’s scores on an examination and students cumulative grade- point average (GPA)

upon graduation?

S-ID 7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID 8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

S-ID 9 Distinguish between correlation and causation.

Interpret Slope:

Interpret y-intercept:

Interpret x-intercept:

2.6 Correlation or Causation & Interpreting Slope & Intercepts Practice 1. Go back and answer all the starred questions in 2.5 (interpret slope & y=intercept,

including even #s) 3. Go back and find the x-intercept (substitute y=0 and solve for x) and interpret for the odd # questions in 2.5 Identify the relationship between the two quantities in the given question as causation or correlation. 4. The number of cold, snowy days and the amount of hot chocolate sold at a ski resort. 5. The number of miles driven and the amount of gas used. 6. The number of additional calories consumed and the amount of weight gained. 7. The age of a child and his/her shoe size. 8. The amount of cars a salesperson sells and how much commission he makes. 9. The number of homework assignments turned in and how well and individual does in class. 10. The annual salary and blood pressure for men ages 20-60. 11. The high divorce rate and lower death rates in the South. Review Problems (complete ALL) 12. Solve: 3𝑥 − 2 + 1 = 13 13. Solve: -2x +5 ≤ 13 14. Evaluate f(-2) if f(x) = -4x -7

15. What is the 23rd term? {(24, 21, 18, 15, …)} Make sure you show the formula used to obtain the answer. 16. Write the equation of the line that passes through (-2, -4), and (4, 2). 17. Graph 2y=4x-6

2.1 resources This page will be removed to cut and add to the 2.1 notes

{-1, 5), (2, 5), (3, 5) (8, 5)}

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