HSF.BF.A.1: Writing Functions That Model a Relationship
In plain English: HSF.BF.A.1 is the Common Core functions standard that asks students to write a function for a relationship between two quantities: an explicit expression, a recursive process or steps for calculation from a context, and new functions built by adding, subtracting, multiplying or dividing standard types. Part c, composing functions, is an advanced (+) standard usually taught in Algebra II or Precalculus.
Write a function that describes a relationship between two quantities.
a.Determine an explicit expression, a recursive process, or steps for calculation from a context.
b.Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
c.(+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Common Core State Standards for Mathematics · Domain: Building Functions (BF) · Cluster: Build a function that models a relationship between two quantities Also written as HSF-BF.A.1 or F-BF.1 · Official standard
Students take a situation described in words and write a function for it. Part a asks for three kinds of rules: an explicit expression such as C(m) = 40 + 25m, a recursive process that says how to get the next value from the current one, and a list of calculation steps that a person would follow with a calculator. Students learn to decide which kind of rule the context suggests and to check it against a few values.
Part b builds new functions by combining familiar ones with arithmetic: profit as revenue minus cost, average cost as total cost divided by quantity, and the official example of a cooling body, where a constant function (room temperature) is added to a decaying exponential (the temperature difference). Part c, an advanced (+) standard, composes functions: the output of one function becomes the input of the next, as in the official weather balloon example T(h(t)).
Learning Objectives
By the end of this lesson, students will be able to:
Write an explicit expression for a relationship between two quantities described in a context, with units and a sensible domain
Describe a context with a recursive process (a starting value and a rule for the next value) and use it to generate values
Turn a list of calculation steps into a single function rule and back
Build a function by adding, subtracting, multiplying or dividing standard functions, and explain what each part means in the model
(+) Compose two functions in context, state what the input and output of the composition are, and evaluate it
Prior Knowledge Required
Students should already be comfortable with:
Function notation and evaluating functions HSF.IF.A.2
Constructing linear functions from a description or a table 8.F.B.4
Linear and exponential functions from a description of growth or decay HSF.LE.A.2
Percent increase and decrease written as multipliers, such as 0.95 for a 5% decrease
Put one situation on the board and ask students to describe the rule in words before writing any symbols.
Warm-Up Prompt
"A parking garage charges $3 for the first hour and $2 for each additional hour, counted in whole hours. How much do 5 hours cost? Explain your steps in words, then write a rule for the cost of h hours."
Collect the different ways students described it. Some will list 3, 5, 7, 9, 11 (a recursive process: add 2 each hour), some will write steps (subtract the first hour, multiply by 2, add 3), and some will write C(h) = 3 + 2(h - 1) = 2h + 1. All three give $11 for 5 hours. Tell students that the standard names exactly these three kinds of rules, and that today they will learn when each is useful.
Direct Instruction25 minutes
Part 1: From a context to a rule (standard a). Model a routine students can reuse on any context:
Name the two quantities and their units: which one is the input and which one is the output?
Find what stays fixed and what changes: a starting amount, a constant rate, a constant percent, a fee.
Choose the kind of rule: explicit when you can jump straight to any input, recursive when the context describes one step at a time, steps for calculation when the context is a procedure.
Write the rule in function notation and state a sensible domain.
Check the rule with two values from the context, and interpret one value in a sentence.
Work through the examples below. For each one, ask students which kind of rule the wording suggests before writing it.
Explicit expression
A gym charges a $40 joining fee and $25 per month. Write the total cost C after m months and find the cost of one year.
Equation: C(m) = 40 + 25m, so C(12) = $340
Recursive process
A patient takes a 200 mg dose every 12 hours. Between doses the body clears 60% of the drug, so 40% remains. Let A(n) be the amount right after dose n.
A store takes 20% off the tag price p, then adds 8% sales tax to the discounted price. Write the steps and a single rule, then find the total for a $45 jacket.
Coffee at 90°C sits in a 20°C room. The difference between the coffee and the room shrinks by 5% each minute. Build T(t) from a constant function and a decaying exponential.
Equation: T(t) = 20 + 70(0.95)ᵗ, so T(10) ≈ 61.9°C and T(t) approaches 20°C
(+) Composing functions (official example c)
Air temperature is T(y) = 15 - 0.0065y °C at height y meters. A weather balloon rises 300 meters per minute from sea level, so h(t) = 300t. Write the temperature at the balloon after t minutes.
Equation: T(h(t)) = 15 - 1.95t, so T(h(20)) = -24°C
Part 2: Combining functions (standard b). Use Diagram 1 to relate each piece of the cooling model to the situation. The constant function 20 is the room temperature, the function 70(0.95)ᵗ is how much warmer than the room the coffee is, and their sum is the coffee's temperature. As t grows, 70(0.95)ᵗ gets close to 0, so T(t) gets close to 20: the coffee never becomes colder than the room. Contrast this with a pure exponential 90(0.95)ᵗ, which would wrongly predict that the coffee cools toward 0°C.
Part 3 (+): Composing functions (standard c). Use Diagram 2. The output of h is a height, and T needs a height as its input, so T(h(t)) makes sense and gives temperature as a function of time. Ask why h(T(y)) makes no sense here: T gives a temperature, and h needs a time. Point out the domain: the lapse-rate model is only valid in the lower atmosphere, below about 11,000 m, so this composition is only valid for about the first 36 minutes.
Guided Practice15-20 minutes
Pairs work through four contexts, one kind of rule each, and compare with another pair before moving on:
Explicit: A 20,000-gallon pool already holds 2,000 gallons and fills at 450 gallons per hour. (V(t) = 2000 + 450t; full when t = 40 hours.)
Recursive: A $1,000 loan charges 2% interest per month, and the borrower pays $100 at the end of each month. (B(0) = 1000, B(n) = 1.02 · B(n - 1) - 100; B(1) = $920, B(2) = $838.40.)
Combining: A coffee cart sells drinks for $4.50 each. Its daily cost is $220 plus $1.30 per drink. (P(x) = 4.5x - (220 + 1.3x) = 3.2x - 220; it needs 69 drinks to make a profit, because 68.75 drinks is not possible.)
(+) Composing: A store gives a 10% member discount, f(x) = 0.9x, and a $10 coupon, g(x) = x - 10. For an $80 purchase, compare f(g(80)) = $63 and g(f(80)) = $62, and say which order the customer should ask for.
Listen for these errors: writing 25 + 40m (rate and fee swapped), forgetting the starting value in a recursive rule, assuming the order of steps never matters (it does not matter for two percent changes, but it does when a flat coupon is involved), and composing in the wrong order.
Independent Practice15 minutes
Students complete four short problems on their own and write the kind of rule in the margin:
An elevator starts 120 m above the ground and descends at 4 m per second. Write H(t) and find when it reaches the ground. (H(t) = 120 - 4t; t = 30 s.)
A ball dropped from 64 inches rebounds to 75% of its previous height on each bounce. Write a recursive rule and find the first three rebound heights. (R(1) = 48, R(n) = 0.75 · R(n - 1); 48, 36, 27 inches.)
A 42°C bath cools in a 20°C bathroom, and the temperature difference shrinks by 3% each minute. Build the model from a constant and an exponential and find the temperature after 10 minutes. (T(t) = 20 + 22(0.97)ᵗ; about 36.2°C.)
(+) For f(x) = x² and g(x) = x - 5, find f(g(7)) and g(f(7)). (4 and 44.)
Closure5-10 minutes
Exit ticket: (1) A bike rental costs $8 plus $4.50 per hour. Write an explicit rule and find the cost of 3 hours. (C(h) = 8 + 4.5h; $21.50.) (2) A pumpkin patch has 40 pumpkins ripe on day 0, and the number ripe grows by 50% each day. Write a recursive rule and find the number ripe on day 3. (P(0) = 40, P(n) = 1.5 · P(n - 1); 135.) (3) (+) If f(x) = 3x and g(x) = x + 2, find f(g(1)). (9.)
Differentiation Strategies
For Struggling Students
Give a two-column table (input, output) for every context and have students fill in three rows before writing any rule
Provide sentence frames: "The output starts at ___ and changes by ___ each ___"
For composition, have students write the middle value in a box first (for example, the height 6000 m) before computing the final output
For Advanced Students
Find the long-run amount of drug in the recursive dosing example by solving A = 0.4A + 200 (about 333 mg) and explain what it means
Write a cooling model for a cold drink warming up in a room and explain why the exponential part is subtracted
Compose three functions: convert the balloon's temperature in °C to °F with F(C) = 1.8C + 32 and write F(T(h(t)))
Assessment Guidance
What to Look For
Check that every rule comes with a clear input, output and units, and that students test it against at least two values from the context. In recursive rules, look for a starting value as well as the step. In combined models, ask students to say in words what each part represents, not only to compute. For composition, the key idea is that the output of the inner function must be a valid input of the outer function; ask students to name the units of the middle quantity.
02
Classroom Activities
3 Activities
1
Three Ways to Write a Rule
20 minPairs
Pairs receive 6 context cards. For each card they write the rule in the form the card asks for, then write the same relationship in a second form and check that both give the same value for one input. The goal is to see explicit rules, recursive processes and calculation steps as different descriptions of one relationship.
Context Cards (with answers)
Card 1 (explicit): Stacked chairs: one chair is 80 cm tall and each extra chair adds 6 cm. H(n) = 80 + 6(n - 1) = 74 + 6n; a stack of 10 is 134 cm
Card 2 (recursive): A culture starts with 50 bacteria and doubles every hour. B(0) = 50, B(t) = 2 · B(t - 1); explicit B(t) = 50 · 2ᵗ; B(5) = 1600
Card 3 (explicit): A $24,000 car loses 15% of its value each year. V(t) = 24000(0.85)ᵗ; V(2) = $17,340
Card 4 (recursive): A savings jar holds $35 and gets $12 each week. M(0) = 35, M(w) = M(w - 1) + 12; explicit M(w) = 35 + 12w; M(8) = $131
Card 5 (steps): To convert °C to °F, multiply by 9/5, then add 32. F(C) = 1.8C + 32; F(25) = 77°F
Card 6 (steps): A $35 phone plan includes 5 GB, and each extra GB costs $10. For g ≥ 5: subtract 5, multiply by 10, add 35. C(g) = 35 + 10(g - 5); C(8) = $65
Discussion Questions
Which cards were easier to describe recursively? What words in the context told you?
Card 6 only works for g ≥ 5. What rule would you use for g < 5?
Why does a recursive rule need a starting value?
Modification for Distance Learning
Share the cards on a slide. Pairs work in a shared spreadsheet: one column for the recursive rule (each cell refers to the cell above) and one for the explicit rule, and they check that the columns match.
2
Build the Cooling Model
25 minGroups of 3-4
Groups work with a table of temperatures for a cup of hot water cooling in a 22°C classroom (invented data, rounded to 0.1°C). They discover that the temperature itself does not decay by a constant factor, but the difference from room temperature does, and they build the model from a constant function plus a decaying exponential, as in the official example for part b.
Data (invented)
t = 0 min: 82.0°C
t = 5 min: 76.0°C
t = 10 min: 70.6°C
t = 15 min: 65.7°C
t = 20 min: 61.4°C
Procedure
Compute the ratios of consecutive temperatures (76.0/82.0 ≈ 0.927, 70.6/76.0 ≈ 0.929): they are not constant
Subtract the room temperature: 60, 54, 48.6, 43.7, 39.4. Now each value is 0.9 times the previous one
Write D(t) = 60(0.9)^(t/5) for the difference and T(t) = 22 + 60(0.9)^(t/5) for the temperature
Graph the data and the model on a graphing tool and explain what the 22, the 60 and the 0.9 mean
Discussion Questions
What temperature does the model predict after a very long time? Why does that make sense?
Why would a model with only an exponential, such as 82(0.93)^(t/5), give a wrong long-term prediction?
How would the model change in a 30°C room?
Challenge Variation
Groups with a thermometer can collect their own data from a cup of warm water (with teacher supervision) and compare their factor with other groups.
3
(+) Composition Chains
20 minPairs
Pairs build compositions from two real rules, name the input, middle and output quantities with units, and decide which order makes sense. This activity covers part c, an advanced (+) standard.
Chains
A stone dropped in a pond makes a circular ripple whose radius grows at 30 cm per second: r(t) = 30t, and the area inside a circle is A(r) = πr². A(r(t)) = 900πt²; after 2 seconds the area is 3600π ≈ 11,310 cm²
A car uses 0.08 liters of fuel per kilometer, F(d) = 0.08d, and travels at 90 km per hour, d(t) = 90t. F(d(t)) = 7.2t liters; a 2.5-hour trip uses 18 liters
Wages: a server earns W(h) = 16h dollars for h hours, and pays 12% in taxes, N(w) = 0.88w. N(W(h)) = 14.08h; a 25-hour week gives $352 after taxes
Procedure
For each chain, draw the arrow diagram input, middle, output, with units
Write the composition in both orders and cross out the one that makes no sense, with a reason
Evaluate the composition at the given input in two ways: step by step and with the single composed rule
Discussion Questions
In the ripple chain, why is the area multiplied by 4 when the time doubles?
What is the domain of F(d(t)) if the tank holds 50 liters?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Cooling Model Built From Two Functions
The coffee model T(t) = 20 + 70(0.95)ᵗ (solid) is the sum of the constant function 20 (the room temperature, dashed) and the decaying exponential 70(0.95)ᵗ (the difference from the room, dotted). Drawn to scale for 0 ≤ t ≤ 60 minutes. The coffee starts at 90°C, is about 61.9°C after 10 minutes, and levels off toward 20°C.
Diagram 2: (+) Composing Height and Temperature
The weather balloon from the official example for part c. The output of h(t) = 300t is a height, which is the input of T(y) = 15 - 0.0065y. The composition T(h(t)) = 15 - 1.95t gives the temperature at the balloon as a function of time.
04
Homework Assignment
~30 min
HSF.BF.A.1 Homework: Writing Functions From Context
Directions: For each problem, name the input and output with units, write the rule in function notation, and check it with one value from the context. Round money to the nearest cent and temperatures to the nearest tenth of a degree.
Part 1: Explicit, Recursive and Step Rules (Problems 1-3)
A water tank holds 1,200 liters and drains at 45 liters per minute. Write an explicit rule V(t) for the volume after t minutes, find when the tank is empty (in minutes and seconds), and state a sensible domain.
A lake is stocked with 2,000 trout. Each year 30% of the trout die or are caught, and then the state adds 500 new trout. Write a recursive process for the number of trout at the end of year n, and use it to find the numbers for years 1, 2 and 3.
A currency booth charges a $5 fee, taken before converting, and then gives 0.92 euros per dollar that remains. Write the calculation steps, then a single rule E(d). How many euros does a traveler get for $300? How many dollars must she bring to get 184 euros?
Part 2: Combining Functions (Problems 4-5)
A bakery's daily cost for x loaves is C(x) = 150 + 1.25x dollars, and it sells each loaf for $4, so R(x) = 4x. (a) Write the profit function P(x) = R(x) - C(x). (b) How many loaves must it sell to make a profit? (c) Write the average cost per loaf A(x) = C(x)/x and find A(100).
A cake comes out of the oven at 175°C and cools in a 20°C kitchen. The difference between the cake and the kitchen shrinks by 10% each minute. (a) Build T(t) by adding a constant function and a decaying exponential, and say what each part means. (b) Find T(10) and T(20). (c) What temperature does the cake approach, and why?
Part 3: (+) Composing Functions (Problem 6)
A factory produces n(t) = 120t phones in t hours, and the cost of producing n phones is C(n) = 2000 + 85n dollars. (a) Write C(n(t)) and simplify it. (b) What are the input and output of C(n(t))? (c) Find the cost of an 8-hour shift. (d) Explain why n(C(t)) makes no sense here.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Setting Up the Rule
Input, output and units named; rule matches the context
Rule mostly correct, units or domain missing
Rule missing or does not fit the context
Recursive and Step Rules
Starting value and step both given; steps match the single rule
Step correct but starting value missing, or one step out of order
Missing or incorrect
Combining and Composing
Correct operation or order; each part interpreted in the context
Correct rule without interpretation
Wrong operation or order
Accuracy and Checking
All values correct and checked against the context
One or two computation errors
Most values incorrect
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. Questions marked (+) cover part c, composing functions.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A plumber charges a $75 service fee plus $60 per hour. Which function gives the cost for h hours of work?
Answer: B
The fee is paid once and the rate is paid for every hour, so C(h) = 75 + 60h. Choice A swaps the fee and the rate. Choice C charges the fee again every hour.
Question 2 of 20 · Multiple Choice
Maya opens a savings account with $150 and deposits $20 every week. Which recursive process describes her balance after n weeks?
Answer: C
A recursive process needs a starting value, a(0) = 150, and a rule for the next value: add 20. Choice D multiplies by 20 each week, which describes growth by a factor, not a deposit. Choice B swaps the starting value and the step.
Question 3 of 20 · Multiple Choice
A lake has 2,500 birds. Each year 4% of the birds leave, and then 50 new birds arrive: a(0) = 2500, a(n) = 0.96 · a(n - 1) + 50. How many birds are there after 2 years?
Answer: A
a(1) = 0.96(2500) + 50 = 2450 and a(2) = 0.96(2450) + 50 = 2402. Choice B stops after one year. Choice C subtracts the 50 birds instead of adding them. Choice D multiplies 2,500 by 0.96 twice and adds both groups of 50 at the end, so the first 50 new birds never lose 4%.
Question 4 of 20 · Multiple Choice
A pizza shop computes an order total in two steps: add a $3 delivery charge to the pizza price p, then add a 15% tip on that sum. Which rule gives the total?
Answer: D
Step 1 gives p + 3. Step 2 multiplies that sum by 1.15, so F(p) = 1.15(p + 3). Choice A applies the tip before adding the delivery charge, so the charge gets no tip. Choice C gives only the tip, not the total.
Question 5 of 20 · Multiple Choice
To convert °F to °C, subtract 32 and then multiply by 5/9. What is 98.6°F in °C?
Answer: D
98.6 - 32 = 66.6, and 66.6 · 5/9 = 37. Choice B multiplies first and then subtracts, doing the steps in the wrong order. Choice C does only the first step, and choice A does only the second step.
Question 6 of 20 · Multiple Choice
A cup of tea cools in a room, and its temperature after t minutes is T(t) = 22 + 68(0.93)ᵗ °C. What does the 22 represent?
Answer: A
As t grows, 68(0.93)ᵗ approaches 0 and T(t) approaches 22, so 22 is the temperature the tea levels off at: the room temperature. Choice B is 22 + 68 = 90, and choice D describes the 7% shrink from the factor 0.93.
Question 7 of 20 · Multiple Choice
For the same model T(t) = 22 + 68(0.93)ᵗ, what is the tea's temperature at t = 0?
Answer: B
(0.93)⁰ = 1, so T(0) = 22 + 68 = 90°C. Choice A is only the difference from the room. Choice D is T(1) = 22 + 68(0.93) = 85.24°C, one minute later.
Question 8 of 20 · Multiple Choice
A food truck's revenue for x meals is R(x) = 12x dollars, and its cost is C(x) = 400 + 7x dollars. Which function gives its profit?
Answer: C
P(x) = R(x) - C(x) = 12x - (400 + 7x) = 5x - 400. Choice A forgets to subtract the whole cost: the minus sign must apply to 400 as well. Choice B adds the cost instead of subtracting it.
Question 9 of 20 · Multiple Choice
A rectangle's length is L(x) = x + 4 and its width is W(x) = 2x, in centimeters. Which function gives its area?
Answer: A
Area is length times width: (x + 4)(2x) = 2x² + 8x. Choice B adds the two functions instead of multiplying. Choice D is the perimeter, 2(3x + 4). Choice C multiplies only the x terms and keeps the 4.
Question 10 of 20 · Multiple Choice
(+) If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?
Answer: B
Work from the inside out: g(3) = 9, then f(9) = 2(9) + 1 = 19. Choice A is g(f(3)) = 7² = 49, the wrong order. Choice C adds f(3) + g(3) = 7 + 9, and choice D multiplies them.
Question 11 of 20 · Multiple Choice
(+) The air temperature is T(y) = 12 - 0.006y °C at height y meters, and a balloon's height is h(t) = 250t meters after t minutes. What is T(h(10))?
Answer: B
h(10) = 2500 meters, and T(2500) = 12 - 0.006(2500) = 12 - 15 = -3°C. Choice A plugs 10 directly into T, as if 10 were a height. Choice C stops at the height. Choice D adds 15 instead of subtracting it.
Question 12 of 20 · Multiple Choice
(+) For the weather balloon in the official example, what are the input and output of T(h(t))?
Answer: D
h takes a time and gives a height; T takes that height and gives a temperature. So T(h(t)) takes a time and gives the temperature at the balloon. Choice C describes h alone.
Question 13 of 20 · Multiple Choice
A 5-gallon water jug holds 640 ounces. Each cup poured uses 8 ounces. Which function gives the ounces left after n cups?
Answer: C
The jug starts at 640 and loses 8 ounces per cup, so W(n) = 640 - 8n. Check: W(80) = 0, and 80 cups of 8 ounces is 640 ounces. Choice A subtracts the first cup twice: it gives 632 when no cups have been poured. Choice B adds water instead of removing it.
Question 14 of 20 · Multiple Choice
A club's field trip costs $480 for the bus plus $15 per student. Which function gives the cost per student when n students go?
Answer: A
Divide the total cost 480 + 15n by the number of students: A(n) = (480 + 15n)/n, which equals 480/n + 15. Choice B divides only the bus cost, then multiplies the ticket cost by n again. Choice D charges the $15 only once for the whole group.
Question 15 of 20 · Short Answer
A candle is 30 cm tall and burns 2.5 cm per hour. Write an explicit function for its height after t hours and find when it burns out. State the domain.
h(t) = 30 - 2.5t. It burns out when 30 - 2.5t = 0, so t = 12 hours. Domain: 0 ≤ t ≤ 12, because the height cannot be negative.
Question 16 of 20 · Short Answer
A gym has 800 members. Each month 5% of the members cancel, and then 60 new members join. Write a recursive process for the number of members after n months and use it to find the number after 2 months.
A tutor's bill is computed in steps: take the number of sessions s, subtract the 2 free trial sessions, then multiply by $35. Write a single rule and find the bill for 10 sessions. For which values of s does your rule make sense?
C(s) = 35(s - 2), so C(10) = 35 · 8 = $280. The rule makes sense for whole numbers s ≥ 2; for 0, 1 or 2 sessions the bill is $0, so the rule would need a separate case for s < 2.
Question 18 of 20 · Short Answer
A cold drink at 4°C sits in a 24°C room, and the difference between the room and the drink shrinks by 6% each minute. Build a model for the drink's temperature by combining a constant function and an exponential function, and find the temperature after 10 minutes.
The difference starts at 24 - 4 = 20 and shrinks: 20(0.94)ᵗ. The drink is that much colder than the room, so T(t) = 24 - 20(0.94)ᵗ. Check: T(0) = 4. T(10) = 24 - 20(0.94)¹⁰ ≈ 13.2°C. The drink warms toward 24°C, the room temperature.
Question 19 of 20 · Short Answer
(+) For f(x) = x - 3 and g(x) = 4x, find f(g(x)) and g(f(x)). Are they the same function?
f(g(x)) = 4x - 3 and g(f(x)) = 4(x - 3) = 4x - 12. They are different: for example, f(g(0)) = -3 but g(f(0)) = -12. The order of composition matters.
Question 20 of 20 · Short Answer
(+) A spherical balloon is being pumped so that its radius is r(t) = 2t cm after t seconds, and the volume of a sphere is V(r) = (4/3)πr³. Write V(r(t)) and find the volume after 3 seconds.
V(r(t)) = (4/3)π(2t)³ = (32/3)πt³. After 3 seconds, r = 6 cm and V = (4/3)π(216) = 288π ≈ 904.8 cm³. The input is time in seconds, and the output is volume in cubic centimeters.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.BF.A.1 mean?
HSF.BF.A.1 asks students to write a function that describes a relationship between two quantities. Part a is about getting the rule from a context, as an explicit expression, a recursive process or a list of calculation steps. Part b builds functions by combining standard functions with +, -, × and ÷. Part c, marked (+), composes functions.
Is HSF.BF.A.1 Algebra 1 or Algebra 2?
Both, depending on the part. Parts a and b usually start in Algebra I with linear and exponential models and continue in Algebra II with more function types. Part c, composition, is marked (+) as additional mathematics for advanced courses and is usually taught in Algebra II or Precalculus.
What is the difference between an explicit and a recursive function?
An explicit rule gives the output directly from the input, such as C(m) = 40 + 25m. A recursive rule gives a starting value and says how to get each value from the one before, such as A(1) = 200, A(n) = 0.4 · A(n - 1) + 200. Explicit rules are faster for a far-away input; recursive rules often match the way a context is described, one step at a time.
What does "steps for calculation" mean in this standard?
It means a procedure a person follows to get the output, such as "take 20% off, then add 8% tax" or "subtract 32, then multiply by 5/9". Students should be able to turn the steps into one rule, such as F(p) = 1.08(0.8p), and to notice when the order of the steps changes the result.
Why does the cooling model add a constant to an exponential?
Because it is the difference between the object and the room that decays exponentially, not the temperature itself. The constant is the room temperature, and the exponential is how much warmer the object still is. Adding them gives a model that levels off at room temperature, which is what really happens. This is the official example for part b.
What is a common mistake when writing a function from a word problem?
A common one is swapping the fixed amount and the rate, for example writing 25 + 40m when the fee is $40 and the monthly charge is $25. Another is dropping the starting value from a recursive rule. Having students check the rule with one or two values from the context catches both.
How do you explain function composition to students?
Describe it as a chain: the output of the first function becomes the input of the second. In the weather balloon example, h turns a time into a height, and T turns that height into a temperature, so T(h(t)) turns a time into a temperature. Asking for the units of the middle quantity helps students choose the correct order.
Does the order of composition matter?
Usually yes. For f(x) = 0.9x (a 10% discount) and g(x) = x - 10 (a $10 coupon), f(g(80)) = 63 but g(f(80)) = 62. In context, one order often makes no sense at all, such as putting a temperature into a function that expects a time.
How is HSF.BF.A.1 tested?
Typical items give a short context and ask students to choose or write the function, to interpret a part of a model such as the 20 in T(t) = 20 + 70(0.95)ᵗ, or to evaluate a composition from rules or tables.
How does HSF.BF.A.1 connect to other standards?
It builds on writing linear functions in grade 8 (8.F.B.4) and on constructing linear and exponential functions (HSF.LE.A.2). It runs alongside sequences (HSF.BF.A.2), which are recursive and explicit rules on the integers. Composition leads to inverse functions (HSF.BF.B.4), where composing a function with its inverse gives back the input.
07
Related Standards
6 standards
These standards connect to HSF.BF.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.B.4Prerequisite
Construct a function to model a linear relationship between two quantities
