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HSF.LE.B.5Common CoreMathFunctionsGrades 9-12

HSF.LE.B.5: Interpreting Parameters of Linear and Exponential Functions

In plain English: HSF.LE.B.5 is the Common Core functions standard that asks students to explain what the numbers in a linear or exponential function mean in a real situation. In f(x) = mx + b, m is a rate with units and b is the starting value; in f(t) = a · bᵗ, a is the initial amount and b is the factor per time unit, such as 1.04 for 4% growth. It is usually taught in Algebra I.

Interpret the parameters in a linear or exponential function in terms of a context.

Common Core State Standards for Mathematics · Domain: Linear, Quadratic, and Exponential Models (LE) · Cluster: Interpret expressions for functions in terms of the situation they model
Also written as HSF-LE.B.5 or F-LE.5 · Official standard
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01

Lesson Plan

65-75 min

Overview

Students explain what each number in a linear or exponential function means in the situation it models. In a linear function f(x) = mx + b, they read m as a rate with units, such as dollars per mile, and b as the value when the input is 0. In an exponential function f(t) = a · bt, they read a as the initial amount and b as the factor for each unit of time, and they convert the factor to a percent rate of growth or decay.

Every function in the lesson comes with a context: taxi fares, gas in a tank, a town's population, caffeine in the body. Students write interpretations in full sentences with units, check them by substituting simple inputs, predict how a change in one parameter changes the situation, and decide when a parameter, such as an intercept outside the model's domain, has no sensible meaning.

Learning Objectives

By the end of this lesson, students will be able to:

  • Interpret the slope and intercept of a linear function in context, with units
  • Interpret the initial value a and the factor b of an exponential function a · bt in context
  • Convert a growth or decay factor to a percent rate per unit of time, and back
  • Interpret parameters in forms such as a · 2t/k, including doubling time and half-life
  • Predict how changing one parameter changes the situation, and judge when a parameter has no sensible meaning

Prior Knowledge Required

Students should already be comfortable with:

  • Finding the rate of change and initial value of a linear function from a description 8.F.B.4
  • Telling linear from exponential situations: equal differences versus equal factors HSF.LE.A.1
  • Converting between percents and decimals, such as 3.5% = 0.035
  • Evaluating expressions with exponents, such as 95(0.87)2

Lesson Procedure

65-75 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Post two functions from everyday life and ask students to write what each number means, in a full sentence with units:

    Warm-Up Prompt

    "A bike-share ride costs C(m) = 0.25m + 1.00 dollars for m minutes. A savings account holds B(t) = 500(1.03)t dollars after t years. What do 0.25, 1.00, 500 and 1.03 each tell you?"

    Collect answers. Many students say "0.25 is the price," so push for units: $0.25 per minute. Students usually see that $1.00 is charged at m = 0 and $500 is the starting balance. The 1.03 is harder: some say "$1.03 per year" or "103%." Leave the question open. Point out that C and B are both functions of time, but their numbers play different roles, and the lesson will name each role.

  2. Direct Instruction20-25 minutes

    Linear functions f(x) = mx + b. The slope m is a rate of change in "output units per input unit," and b is the output when the input is 0. Use Diagram 1. Exponential functions f(t) = a · bt. The parameter a is the value at t = 0, and b is the factor for each one-unit increase in t. If b > 1, the growth rate is b - 1 as a percent; if 0 < b < 1, the decay rate is 1 - b. In a · bt/k, the quantity is multiplied by b every k units of time. Give students four questions to ask about any parameter:

    1. Name the input and output with units: for example, t in hours and C in milligrams.
    2. Identify the form: a constant added each unit (linear) or a constant factor each unit (exponential).
    3. Say what each parameter measures, with units: a starting value, a rate per unit, or a factor per unit (and its percent).
    4. Check the sentence against the function: substitute t = 0 and t = 1 to confirm your interpretation.
    5. Ask whether it makes sense: is t = 0 inside the domain where the model is valid? Is the sign reasonable?

    • Linear, increasing

      A taxi fare is F(d) = 2.75d + 3.50 dollars for a ride of d miles.

      Equation: 2.75: the fare rises $2.75 per mile. 3.50: a $3.50 base fare charged at d = 0.

    • Linear, decreasing

      On a highway trip, a car has G(h) = 14 - 2.2h gallons of gas left after h hours.

      Equation: 14: the tank holds 14 gallons at the start. -2.2: the car uses 2.2 gallons per hour.

    • Exponential growth

      A town's population is P(t) = 18,400(1.035)t, t years after 2020.

      Equation: 18,400: the 2020 population. 1.035: the population grows 3.5% per year.

    • Exponential decay

      After a cup of coffee, C(h) = 95(0.87)h mg of caffeine remain in the body after h hours.

      Equation: 95: the milligrams at h = 0. 0.87: 87% remains each hour, so 13% leaves per hour.

    • Exponential, time in the exponent

      A bacteria culture has N(t) = 250 · 2t/20 cells after t minutes.

      Equation: 250: the starting number of cells. 2 and 20: the culture doubles every 20 minutes.

    After the gas example, ask how long the gas lasts (14/2.2 ≈ 6.4 hours) and note that the model only works while the car is driving. After the caffeine example, use Diagram 2: 95 × 0.87 = 82.65 after one hour, and about half is left after 5 hours. Then return to the warm-up: 1.03 means the balance grows 3% per year.

  3. Guided Practice15 minutes

    Pairs interpret every parameter in three models, then answer one follow-up each: a pool with V(h) = 18,000 - 1,500h gallons after h hours of draining (18,000 gallons at the start, 1,500 gallons drained per hour; empty after 12 hours); a streaming service with S(t) = 3,200 + 150t subscribers after t months (3,200 at launch, 150 new subscribers per month); and an apartment with rent R(y) = 1,650(1.04)y dollars after y years ($1,650 now, rising 4% per year; $1,716 next year). For the rent, ask: "What would change if 1.04 became 1.06?" (The rent would rise 6% per year: $1,749 next year. The $1,650 stays the same.) Listen for "per" and units in every answer, and for students who describe 1.04 as a 104% increase.

  4. Independent Practice15 minutes

    Students interpret both parameters in four functions and check each interpretation by substituting 0 and 1: a seedling h(d) = 0.8d + 4.5 cm tall after d days (4.5 cm when planted, 0.8 cm of growth per day); a phone battery B(h) = 100 - 12.5h percent after h hours of video (full at the start, 12.5 percentage points used per hour, empty after 8 hours); a video with V(d) = 1,200(1.35)d views after d days (1,200 on day 0, 35% more each day); and a used bike worth W(y) = 640(0.8)y dollars after y years ($640 now, losing 20% of its value per year).

  5. Closure5-10 minutes

    Exit ticket: (1) After a snowfall, the snow depth is S(h) = 30 - 1.5h centimeters, h hours after the sun comes out. Interpret 30 and -1.5. (30 cm at the start; it melts 1.5 cm per hour.) (2) A town's population is P(t) = 12,000(0.97)t. Interpret 12,000 and 0.97. (12,000 people now; the population falls 3% per year.) (3) In one sentence, explain how a slope and a growth factor are different.

Differentiation Strategies

For Struggling Students

  • Give sentence frames: "The ___ is ___ units when ___ is 0" and "Each ___, the ___ changes by ___ units" (or "is multiplied by ___")
  • Have students make a three-row table for t = 0, 1, 2 before interpreting, so they can see the starting value and the difference or factor
  • Use a percent strip: 100% in the middle, 1.xx as "keeps 100%, adds xx%" and 0.xx as "keeps xx%, loses the rest"

For Advanced Students

  • Ask students to rewrite 5,000 · 2t/9 with a yearly factor (about 1.080) and interpret both forms (extension toward HSF.IF.C.8)
  • Give a linear model whose intercept is outside the domain, such as a model that is valid only for inputs from 3 to 10, and ask for a short argument about why it has no meaning
  • Ask students to write a linear and an exponential model that agree at t = 0 and t = 1, and interpret how the parameters differ

Assessment Guidance

What to Look For

Listen for units and the word "per" in every rate: "2.75 dollars per mile," not "2.75 is the slope." For exponential functions, check that students convert the factor correctly: 1.06 is 6% growth and 0.93 is 7% decay, not 93%. Ask students to test each interpretation by substituting t = 0 and t = 1. When the intercept falls outside the domain of the model, look for students who say so instead of forcing a meaning.

02

Classroom Activities

3 Activities

1

Parameter Match

20 minPairs

Pairs match 8 function cards with 8 context cards. Three pairs of cards differ only in which number plays which role, so students have to interpret each parameter, not just spot a familiar number.

The 8 Matches (answers for the teacher)

  • C(n) = 15n + 40: a $40 sign-up fee plus $15 per class
  • C(n) = 40n + 15: a $15 sign-up fee plus $40 per class
  • H(t) = 1,500 - 40t: a hot-air balloon at 1,500 feet descending 40 feet per minute
  • A(t) = 40(1.15)t: 40 newsletter subscribers, growing 15% per week
  • A(t) = 40(0.85)t: 40 grams of a substance, 15% of which breaks down each day
  • A(t) = 15(1.40)t: a 15 cm² patch of mold growing 40% per day
  • A(t) = 1,500 · 2t/10: 1,500 bacteria doubling every 10 hours
  • A(t) = 1,500(0.5)t/10: a 1,500 mg sample with a half-life of 10 hours

Procedure

  • Pairs match all 8 cards, then write one sentence on each function card explaining the parameter that decided the match
  • For each near-miss pair (the two class plans, the two 15% cards, the two 1,500 cards), pairs substitute t = 1 or n = 1 to confirm
  • Two pairs compare answers and settle any disagreement

Discussion Questions

  • The two class plans cost the same for 1 class. For which numbers of classes is each plan cheaper?
  • How can two cards contain "15%" but use factors 1.15 and 0.85?
2

Change One Number

20 minGroups of 4

Each group gets two base models: a gym plan G(m) = 25m + 60 dollars for m months and a savings account B(t) = 800(1.05)t dollars after t years. Each member changes one parameter, predicts what happens in the context and on the graph, and checks the prediction with a graphing tool.

Role Cards

  • Member 1: change 25 to 30. (Each month costs $5 more; a year costs $420 instead of $360; the line is steeper.)
  • Member 2: change 60 to 90. (A bigger joining fee; a year costs $390; the line moves up but stays parallel.)
  • Member 3: change 1.05 to 1.08. (8% growth per year; after 10 years about $1,727.14 instead of $1,303.12; the curve rises faster.)
  • Member 4: change 800 to 1,000. (A bigger first deposit, still 5% per year; after 10 years about $1,628.89; every value grows by the same factor, 1.25.)

Procedure

  • Each member writes a prediction in words before graphing
  • The group graphs the base model and all four changed models in one window, using a different color for each
  • Each member reports: "Changing ___ changes ___ in the situation, and the graph ___"

Modification for Distance Learning

Share one graph link per group. Each member adds a slider for their parameter, moves it and records one sentence in the shared document.

3

Headline Check

15 minPairs

Pairs read 6 invented news headlines, each with the function it claims to describe. They decide whether the headline interprets the parameters correctly and rewrite the ones that do not.

Headlines (answers for the teacher)

  • "Cab fares: $4 to start, $2.40 per mile" for F(d) = 2.40d + 4 (correct)
  • "Town grows 1.02% per year" for P(t) = 9,500(1.02)t (wrong: 2% per year)
  • "Laptops lose 80% of their value every year" for V(t) = 1,100(0.80)t (wrong: they lose 20% and keep 80%)
  • "Reservoir at 850 million gallons, dropping 6 million gallons per week" for R(w) = 850 - 6w (correct)
  • "Followers double every 4 months" for F(m) = 250 · 2m/4 (correct)
  • "Bank adds $1.035 every year" for B(t) = 2,000(1.035)t (wrong: 3.5% per year)

Procedure

  • Pairs mark each headline correct or wrong and justify with a substitution, such as P(1) = 9,690
  • Pairs rewrite every wrong headline so that it names the parameter, its units and its meaning
  • The class votes on the clearest rewrite for each wrong headline

Challenge Variation

Pairs write their own function and a subtly wrong headline for it, then trade with another pair to find and fix the error.

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: Slope and Intercept of a Taxi Fare

0 1 2 3 4 5 6 7 8 0 4 8 12 16 20 24 28 d (miles) F ($) +1 mile +$2.75 (0, 3.50) (4, 14.50) (5, 17.25) F(d) = 2.75d + 3.50 Slope 2.75: the fare rises $2.75 for each mile driven Intercept 3.50: the fare is $3.50 before any miles (d = 0)
The graph of F(d) = 2.75d + 3.50 drawn to scale for rides of 0 to 8 miles. The line starts at (0, 3.50), the base fare, and every extra mile adds $2.75, shown by the step from (4, 14.50) to (5, 17.25).

Diagram 2: Initial Value and Decay Factor of Caffeine

0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 h (hours after drinking) C (mg) 95 82.65 71.91 47.35 about half after 5 hours C(h) = 95(0.87)h a = 95: 95 mg at h = 0 b = 0.87: each hour, 87% remains, so 13% leaves the body 82.65 = 95 × 0.87
The graph of C(h) = 95(0.87)h drawn to scale for 0 to 10 hours. The curve starts at 95 mg, and each hour the amount is multiplied by 0.87: 95, 82.65, 71.91 and so on. After 5 hours about 47.35 mg, roughly half, remain.

04

Homework Assignment

~30 min

HSF.LE.B.5 Homework: Interpreting Parameters in Context

Directions: For each function, interpret every parameter in a complete sentence with units. Then answer the question that follows. Check each interpretation by substituting a simple input such as 0 or 1.

Part 1: Linear Functions (Problems 1-3)

  1. A plumber charges C(h) = 85h + 60 dollars for a job that takes h hours. Interpret 85 and 60, then find the cost of a 3-hour job.
  2. A hiker descending a mountain is at elevation E(t) = 2,450 - 8t meters after t minutes. Interpret 2,450 and -8. When will the hiker reach the trailhead at 1,850 meters?
  3. At a pizza shop, the price of a pizza with diameter d inches is P(d) = 1.25d - 2.50 dollars, for 10 ≤ d ≤ 18. Interpret 1.25. Explain why -2.50 has no sensible meaning here. Find the price of a 14-inch pizza.

Part 2: Exponential Functions (Problems 4-6)

  1. A city's population is P(t) = 64,000(1.018)t, t years from now. Interpret 64,000 and 1.018, including the percent rate. Estimate the population in 10 years.
  2. A smartphone's resale value is V(t) = 950(0.68)t dollars after t years. Interpret 950 and 0.68, including the percent rate. Find the value after 2 years.
  3. Algae in a pond cover A(d) = 12 · 2d/6 square meters after d days. (a) Interpret 12, 2 and 6. (b) A second pond follows L(d) = 12 + 4d. Interpret 12 and 4 in that model. (c) Which pond has more algae after 18 days?

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Linear ParametersRate and starting value correct, with units and "per"Correct meaning, units missingRate and intercept confused
Exponential ParametersInitial value and factor correct, percent rate correctFactor correct, percent rate wrongFactor read as a linear rate
ReasonablenessNotes domain limits and meaningless interceptsMentions the issue without a reasonNot addressed
ComputationAll values correct and rounded sensiblyOne errorSeveral errors

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Choose the interpretation that fits the context, with the right units. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    A gym charges C(m) = 35m + 50 dollars for m months of membership. What does 50 represent?

  2. Question 2 of 20 · Multiple Choice

    A candle's height is h(t) = 24 - 1.5t centimeters after burning for t hours. What does -1.5 mean?

  3. Question 3 of 20 · Multiple Choice

    A savings account balance is A(t) = 2,000(1.045)t dollars after t years. What does 1.045 tell you?

  4. Question 4 of 20 · Multiple Choice

    A truck's value is V(t) = 32,000(0.91)t dollars after t years. By what percent does its value decrease each year?

  5. Question 5 of 20 · Multiple Choice

    The number of cells in a lab culture is N(t) = 40 · 3t/5, with t in hours. What does the 5 mean?

  6. Question 6 of 20 · Multiple Choice

    An oven's temperature is T(m) = 70 + 25m degrees Fahrenheit, m minutes after it is turned on (until it reaches 350°F). What does 70 represent?

  7. Question 7 of 20 · Multiple Choice

    A school club's profit from selling n T-shirts is P(n) = 4n - 180 dollars. What does -180 represent?

  8. Question 8 of 20 · Multiple Choice

    A town of 5,000 people is losing 6% of its population each year. Which function models the population after t years?

  9. Question 9 of 20 · Multiple Choice

    Two savings plans start in the same year: Plan A is A(t) = 600 + 50t and Plan B is B(t) = 600(1.05)t, in dollars after t years. Which statement is true?

  10. Question 10 of 20 · Multiple Choice

    The mass of a radioactive substance is f(t) = 1,200(0.5)t/8 milligrams after t days. What is its half-life?

  11. Question 11 of 20 · Multiple Choice

    During a storm, the snow depth is D(h) = 18 + 1.2h inches, h hours after noon. Which is the best interpretation of 1.2?

  12. Question 12 of 20 · Multiple Choice

    A loan balance grows as B(m) = 3,000(1.004)m dollars after m months. What is the monthly percent growth rate?

  13. Question 13 of 20 · Multiple Choice

    In P(t) = 450(1.07)t, the number of members of a club after t years, the 450 is changed to 600. What does this change mean in the context?

  14. Question 14 of 20 · Multiple Choice

    A driver is d(t) = 65t + 120 miles from home after t more hours on the highway. If the driver drove faster, which parameter would change?

  15. Question 15 of 20 · Short Answer

    A tutoring service charges C(h) = 45h + 20 dollars for h hours. Interpret 45 and 20 in this context, then find the cost of 4 hours.

  16. Question 16 of 20 · Short Answer

    The number of invasive fish in a lake is modeled by F(t) = 250(1.12)t, t years after they were first counted. Interpret 250 and 1.12, then estimate the number of fish after 3 years.

  17. Question 17 of 20 · Short Answer

    The amount of a medicine in a patient's blood is M(t) = 80(0.75)t mg, t hours after a dose. Interpret 80 and 0.75, then find the amount after 2 hours.

  18. Question 18 of 20 · Short Answer

    During a drought, the water depth of a reservoir is D(t) = 42 - 0.6t meters after t weeks. Interpret 42 and -0.6. The model predicts D = 0 after 70 weeks. Should you trust that prediction? Explain.

  19. Question 19 of 20 · Short Answer

    A newsletter's subscribers (in thousands) after t months could be modeled by S(t) = 3 + 1.2t or by E(t) = 3(1.2)t. Explain what 3 and 1.2 mean in each model and how the two situations differ.

  20. Question 20 of 20 · Short Answer

    An investment is worth V(t) = 5,000 · 2t/9 dollars after t years. Interpret 5,000, 2 and 9, then find the value after 27 years.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does HSF.LE.B.5 mean?

It means students can explain what each number in a linear or exponential function says about the situation. In a taxi fare F(d) = 2.75d + 3.50, the 2.75 is dollars per mile and the 3.50 is the base fare. In C(h) = 95(0.87)h, the 95 is the starting amount and 0.87 means 13% leaves each hour.

Is HSF.LE.B.5 Algebra 1 or Algebra 2?

It is usually taught in Algebra I, alongside writing linear and exponential functions (HSF.LE.A.2). It comes back in Algebra II with more complex exponential forms and with logarithms (HSF.LE.A.4).

What is a parameter in a function?

A parameter is a number that stays fixed for a given situation, such as m and b in f(x) = mx + b or a and b in f(t) = a · bt. The variable (x or t) changes; the parameters describe the situation. Changing a parameter describes a different situation, such as a different fare or a different growth rate.

How do you interpret slope and y-intercept in a word problem?

The slope is the change in output for each one-unit increase in input, with units "output per input," and the y-intercept is the output when the input is 0. For G(h) = 14 - 2.2h gallons after h hours, the car uses 2.2 gallons per hour and starts with 14 gallons.

How do you find the percent growth or decay rate from an exponential function?

Compare the factor b with 1. If b > 1, the growth rate is b - 1: 1.035 means 3.5% growth per unit of time. If 0 < b < 1, the decay rate is 1 - b: 0.87 means 13% decay. The factor is what remains after one unit; the rate is what is added or lost.

What does the number in the exponent mean, as in 2^(t/20)?

It tells you how long it takes for the quantity to be multiplied by the base once. In N(t) = 250 · 2t/20, the exponent grows by 1 every 20 minutes, so the culture doubles every 20 minutes. In (0.5)t/30, the quantity halves every 30 units: a half-life of 30.

What if the intercept does not make sense in the context?

Say so and explain why. A model is only valid on a certain domain. In H(a) = 6a + 77, the height in centimeters of children aged 3 to 10, age 0 is not in the domain, so 77 is not a birth length (babies are about 50 cm long). It is just a number that places the line. Students should not invent a meaning for it.

What are common mistakes when interpreting parameters?

A common one is reading the factor as the rate: calling 0.82 an "82% decrease" instead of an 18% decrease. Others are leaving out units, reading the slope of a linear model as a percent, and confusing a parameter with an input value, such as saying 3.50 in F(d) = 2.75d + 3.50 is "3.50 miles."

How is HSF.LE.B.5 tested on the SAT?

Questions that ask for the best interpretation of a number in a linear or exponential model appear on the SAT, in the Algebra and Advanced Math domains. A typical question gives a function with a context and asks what one parameter represents. Answer choices often include the rate and the starting value swapped, so units help.

How does this connect to statistics?

In statistics, students fit a line to data and interpret its slope and intercept in context (HSS.ID.C.7). The skill is the same, with one addition: the fitted slope describes a trend in the data, so interpretations say "is predicted to" or "on average."