HSF.TF.B.5: Modeling Periodic Phenomena with Trigonometric Functions
In plain English: HSF.TF.B.5 is the Common Core functions standard that asks students to choose a sine or cosine function to model a repeating situation, such as a Ferris wheel, the tides or a sound wave, with a given amplitude, frequency and midline. Students write y = A sin(Bt) + D or y = A cos(Bt) + D and explain what each number means in the context. It is a modeling standard usually taught in Algebra II or Precalculus.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Common Core State Standards for Mathematics · Domain: Trigonometric Functions (TF) · Cluster: Model periodic phenomena with trigonometric functions Also written as HSF-TF.B.5 or F-TF.5 · Official standard
Students learn to choose a trigonometric function that models a repeating situation when the amplitude, frequency and midline are given or can be read from the context. They write models of the form y = A sin(Bt) + D or y = A cos(Bt) + D, where |A| is the amplitude, y = D is the midline, and B = 2π × frequency = 2π/period. They also decide between sine and cosine, and the sign of A, from what happens at t = 0.
Contexts include a Ferris wheel, tides, a tuning fork, household voltage and a pendulum students build in class. Throughout, students state units, check the model at a few key times, and explain what each parameter means in the situation, including when a model gives a value that does not fit the context.
Learning Objectives
By the end of this lesson, students will be able to:
Find the amplitude and the midline of a periodic situation from its maximum and minimum values
Convert between frequency and period and find B = 2π × frequency for a model in the given time unit
Choose sine or cosine, and the sign of A, to match the behavior of the situation at t = 0
Write and check a trigonometric model with a specified amplitude, frequency and midline
Interpret the amplitude, frequency and midline of a model in the context, with units
Prior Knowledge Required
Students should already be comfortable with:
Sine and cosine of real numbers from the unit circle HSF.TF.A.2
Periodicity of sine and cosine with period 2π HSF.TF.A.4
Key features of graphs: maximum, minimum and intervals where a function increases HSF.IF.B.4
Vertical stretches and shifts of graphs, f(x) + k and k f(x) HSF.BF.B.3
Draw a clock on the board and give the measurements before posting the question:
Warm-Up Prompt
"The minute hand of a wall clock is 10 cm long, and the center of the clock is 150 cm above the floor. Sketch the height of the tip of the minute hand above the floor from 12:00 to 2:00. What are the highest and lowest heights, and how long does it take for the pattern to repeat?"
Students should find a highest height of 160 cm (at 12:00 and 1:00), a lowest height of 140 cm (at 12:30 and 1:30), and a repeat time of 60 minutes. Ask what height the tip is at "on average" (150 cm, the height of the center) and how far above and below that it goes (10 cm). Tell students that these three facts are the midline, the amplitude and the period, and that today they will turn them into an equation.
Direct Instruction25 minutes
Introduce the two model forms, y = A sin(Bt) + D and y = A cos(Bt) + D, and build the recipe for choosing each number. Use Diagram 1 for the first example and Diagram 2 for the sine or cosine choice.
Midline: D is the average of the maximum and the minimum, D = (max + min)/2. The midline is the horizontal line y = D.
Amplitude: |A| is the distance from the midline to the maximum, |A| = (max - min)/2. It is always positive and has the units of the output.
Frequency and period: the frequency f is the number of cycles per unit of time and the period is the time for one cycle, so f = 1/period. Since sine and cosine complete one cycle over 2π, B = 2πf = 2π/period. The unit of t must match the unit in the frequency.
Sine or cosine: look at t = 0. On the midline and rising: A sin(Bt) + D. At the maximum: A cos(Bt) + D. At the minimum: -A cos(Bt) + D. On the midline and falling: -A sin(Bt) + D.
Check: substitute t = 0, a quarter period and half a period. The outputs should be the starting value, a midline or extreme value, and the value half a cycle later.
Ferris wheel, starting at the bottom
A Ferris wheel is 50 m in diameter, its center is 28 m above the ground, and it turns once every 12 minutes. A rider boards at the lowest point. Write the rider's height h in meters after t minutes.
Equation: Amplitude 25, midline h = 28, frequency 1/12 turn per minute, B = 2π/12 = π/6: h(t) = 28 - 25 cos(πt/6)
Tides, starting at high tide
At a harbor, high tide is 3.1 m and low tide is 0.5 m, and high tides are 12.4 hours apart. Write the water height h after t hours if t = 0 is a high tide.
A tuning fork sounds the note A at 440 hertz (cycles per second). Model the change in air pressure p, in pascals, with amplitude 0.02 Pa and midline p = 0, starting at the midline and rising.
Equation: B = 2π(440) = 880π: p(t) = 0.02 sin(880πt), t in seconds
Parameters given directly
Write a function with amplitude 3, midline y = -1 and frequency 2 cycles per unit that starts on its midline and rises.
Equation: B = 2π(2) = 4π: y = 3 sin(4πx) - 1
Reading the parameters back
State the amplitude, midline, period and frequency of y = -4 cos(πx/3) + 7, and describe where it starts.
Equation: Amplitude 4, midline y = 7, period 2π/(π/3) = 6, frequency 1/6; it starts at its minimum, y = 3
After Example 3, stress the difference between the frequency f (440 cycles per second) and the coefficient B (880π radians per second). Many students write sin(440t), which has a frequency of 440/(2π), about 70 cycles per second. Going further: when t = 0 is not at a maximum, minimum or midline crossing, a horizontal shift, y = A cos(B(t - h)) + D, is needed. That is an extension of this standard and is not assessed here.
Guided Practice15 minutes
Pairs write a model for each situation and check it at t = 0 and a quarter period. (1) A buoy moves between 0.6 m below and 0.6 m above its rest position, completing one cycle every 5 seconds, and starts at rest position moving up: y = 0.6 sin(2πt/5). (2) In a city, the average daily high is 88°F in July and 52°F in January. With t = 0 in January and t in months, T = 70 - 18 cos(πt/6). (3) Household voltage in the United States alternates at 60 hertz with an amplitude of about 170 volts and midline 0, starting at 0 and rising: V = 170 sin(120πt). Listen for these errors: using the maximum as the amplitude, using the frequency as B, and choosing cosine when the situation starts on the midline. For (2), ask whether the model predicts July correctly (t = 6 gives 88°F) and whether a monthly temperature really follows a perfect sine curve.
Independent Practice15 minutes
Students work alone. (1) Give the amplitude, midline, period and frequency of y = 6 sin(πx/5) + 2. (Answer: 6, y = 2, 10, 1/10.) (2) Write a function with amplitude 0.5, midline y = 4 and period 3 that starts at its maximum. (Answer: y = 0.5 cos(2πx/3) + 4.) (3) The hub of a wind turbine is 80 m above the ground, each blade is 40 m long, and the rotor turns 15 times per minute. Write the height of a blade tip after t minutes if it starts at the top. (Answer: h = 40 cos(30πt) + 80.) (4) Which has the greater frequency, y = 2 sin(3x) or y = 2 sin(6x), and by what factor? (Answer: the second, twice as great.)
Closure5-10 minutes
Exit ticket: (1) Write a function with amplitude 5, midline y = 12 and frequency 4 cycles per second that starts at its minimum. (Answer: y = -5 cos(8πt) + 12.) (2) What is the frequency of y = 9 cos(πt/4) + 1? (Answer: 1/8 cycle per unit, since the period is 8.) (3) In one sentence, explain the difference between the frequency and B.
Differentiation Strategies
For Struggling Students
Give a four-row organizer (midline, amplitude, B, sine or cosine) and have students fill one row at a time before writing the equation
Start with situations where the midline is 0 and the period is 2π, then change one parameter at a time
Have students mark the maximum, minimum and midline on a quick sketch before any computation
For Advanced Students
Give a situation where t = 0 is between key points, such as a rider boarding 3 minutes after passing the bottom, and ask for a model with a horizontal shift (an extension)
Ask students to explain why a sine model and a cosine model can describe the same situation, and to convert one into the other
Ask students to compare a pendulum model with data that shows the swing slowly getting smaller, and to describe what the model leaves out
Assessment Guidance
What to Look For
Check that students compute D and |A| from the maximum and minimum and do not confuse the amplitude with the maximum. For frequency, look for B = 2π × frequency with matching time units, and for a stated period as a check. Students should justify sine or cosine, and the sign of A, from the behavior at t = 0. In context problems, ask for units on every parameter and for an interpretation such as "the rider is never lower than 3 m" or "high tides come about every 12.4 hours".
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Classroom Activities
3 Activities
1
Pendulum Lab
25 minGroups of 3
Groups build a pendulum from string and washers, measure its amplitude and period, and write a cosine model for the horizontal position of the bob. They then change the length and see how the frequency changes while the amplitude stays their choice.
Procedure
Tape a string to the edge of a table so that 1.0 m hangs below the tape, and tie three washers to the end
Pull the bob 15 cm to the right of its rest position and release it. One student times 10 full swings (right, left and back to the right)
Divide by 10 to get the period. A 1.0 m pendulum has a period of about 2.0 seconds, a frequency of about 0.5 swing per second
With the period 2.0 s, write the position x in cm to the right of rest after t seconds: x(t) = 15 cos(πt)
Shorten the string to 0.25 m and repeat. The period should be about 1.0 second, so x(t) = 15 cos(2πt)
Discussion Questions
Why is cosine the natural choice here? What would change if you started timing as the bob passed through the rest position moving right?
Why does timing 10 swings give a better period than timing one swing?
After a minute, the swing is smaller than 15 cm. Which parameter of the model is no longer accurate, and what does the model leave out?
Modification for Distance Learning
Students use a free online pendulum simulation, set the length and the starting angle, and read the period from the simulation's timer. They post their model and one screenshot to the class discussion board.
2
Model Match
20 minPairs
Pairs match 8 description cards to 8 equation cards. The cards come in near-miss pairs that differ in only one parameter, so students have to read amplitude, frequency, midline and starting point carefully.
The 8 Matches
Amplitude 2, midline y = 0, one cycle every 4 seconds, starts at the maximum: y = 2 cos(πt/2)
Amplitude 2, midline y = 0, 4 cycles per second, starts at the maximum: y = 2 cos(8πt)
Amplitude 4, midline y = 2, period 2, starts on the midline rising: y = 4 sin(πt) + 2
Amplitude 2, midline y = 4, period 2, starts on the midline rising: y = 2 sin(πt) + 4
Amplitude 3, midline y = 10, period 8, starts at the minimum: y = -3 cos(πt/4) + 10
Amplitude 3, midline y = 10, period 8, starts on the midline falling: y = -3 sin(πt/4) + 10
Amplitude 5, midline y = -1, frequency 1/6, starts at the maximum: y = 5 cos(πt/3) - 1
Amplitude 5, midline y = -1, frequency 6, starts at the maximum: y = 5 cos(12πt) - 1
Procedure
Shuffle both sets. Partners take turns choosing a description card and finding its equation
For each match, the other partner checks one value, such as y at t = 0, and the period
When all 8 are matched, pairs write one sentence for each near-miss pair naming the parameter that differs
Challenge Variation
Pairs write two new near-miss pairs of their own, using a real context for each, and trade them with another pair.
3
Design a Ride
20 minGroups of 3-4
Groups design a Ferris wheel within given limits, write the model for a rider who boards at the bottom, and trade designs with another group, which must write the model from the description alone.
Design Limits
Diameter between 30 m and 80 m
Lowest point of the wheel between 2 m and 5 m above the ground
One full turn takes between 8 and 20 minutes
Procedure
Choose the diameter, the height of the lowest point and the time for one turn. Write them on a card as a short description
Write the model h(t) and graph two full turns on graph paper, labeling the midline, amplitude and period
Trade description cards with another group. Each group writes the other group's model and compares it with the original
Sample design: diameter 64 m, lowest point 4 m, one turn every 16 minutes gives h(t) = 36 - 32 cos(πt/8)
Discussion Questions
Which design choice changes the midline? Which changes the amplitude? Which changes B?
If two designs have the same diameter but one turns twice as fast, how do their models differ?
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Diagrams & Visual Aids
2 diagrams
Diagram 1: A Ferris Wheel Model with Its Amplitude, Period and Midline
Graph of h(t) = 28 - 25 cos(πt/6), the height of a rider on a 50 m wheel whose center is 28 m high and which turns once every 12 minutes, drawn to scale for two turns. The rider boards at the minimum, 3 m, and reaches the top, 53 m, after 6 minutes.
Diagram 2: Choosing Sine or Cosine from the Starting Point
The behavior at t = 0 decides the function. Starting on the midline and rising calls for sine, starting at a maximum calls for cosine, and a negative sign in front flips the graph over the midline.
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Homework Assignment
~30 min
HSF.TF.B.5 Homework: Writing Trigonometric Models
Directions: For every model, show how you found the midline, the amplitude and B, and say why you chose sine or cosine. Give units. Check each model at t = 0 and at one more time.
Part 1: Writing Models from Given Features (Problems 1-2)
Write a function with amplitude 6, midline y = 10 and frequency 1/4 cycle per unit that starts at its maximum. State its period.
The water depth at the end of a pier is 6.2 m at high tide and 2.6 m at low tide, and high tides are 12.5 hours apart. Write the depth d after t hours if t = 0 is a low tide. What does your model give at t = 3.125 and at t = 6.25? Explain both values.
Part 2: Models in Context (Problems 3-5)
Middle C on a piano has a frequency of about 262 hertz. Write a model for the change in air pressure with amplitude 0.05 Pa and midline 0, starting at 0 and rising. Find the period in seconds, rounded to 4 decimal places.
A riverboat's paddle wheel has a radius of 4 m, its axle is 3 m above the water, and it turns once every 10 seconds. A point on the rim starts at the top. Write the point's height h above the water after t seconds, and state the amplitude, midline and frequency. What does the minimum of your model mean?
A weight on a spring bobs between 12 cm and 20 cm above a table and makes 3 complete bounces every 2 seconds. It starts at its highest point. Write its height h after t seconds.
Part 3: Finding the Error (Problem 6)
A student wants a model with amplitude 8, midline y = 2 and frequency 3 cycles per second, starting on the midline and rising, and writes y = 8 sin(3t) + 2. What is the frequency of the student's function? Write the correct model.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Midline and Amplitude
Both found from the maximum and minimum and stated with units
One of the two correct
Both missing or incorrect
Frequency and B
B = 2π × frequency with matching time units, period stated
Correct period but B wrong, or units mismatched
Frequency used as B or missing
Function Choice
Sine or cosine and sign of A match t = 0, with a reason
Correct choice without a reason
Choice does not match the situation
Interpretation
Checks and context questions answered with units
Checks done, interpretation incomplete
No checks or interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Use radian mode on any calculator. Choose an answer to see whether it is right and why. Your score updates as you go, and Reset quiz clears every answer.
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
What is the amplitude of y = -5 sin(2x) + 3?
Answer: B
The amplitude is |A| = |-5| = 5, the distance from the midline y = 3 to the maximum 8. Choice A gives A itself, but an amplitude is a distance and is never negative; the negative sign only flips the graph. Choice C is the midline, and choice D is B.
Question 2 of 20 · Multiple Choice
What is the midline of y = 4 cos x - 6?
Answer: C
The constant term D = -6 shifts the graph down 6, so the midline is y = -6, halfway between the maximum -2 and the minimum -10. Choice A uses the amplitude. Choice B is the maximum value.
Question 3 of 20 · Multiple Choice
What is the period of y = sin(πx/7)?
Answer: D
Period = 2π/B = 2π/(π/7) = 14. Choice A is B itself. Choice B is 7, from dividing π by π/7. Choice C forgets that π cancels: 2π ÷ (π/7) = 2 × 7 = 14.
Question 4 of 20 · Multiple Choice
What is the frequency, in cycles per unit of t, of y = 3 cos(10πt)?
Answer: A
Frequency = B/(2π) = 10π/(2π) = 5 cycles per unit. Choice B is B, the coefficient, not the number of cycles per unit. Choice C is the period, 1/5, the reciprocal of the frequency.
Question 5 of 20 · Multiple Choice
A quantity varies between a maximum of 17 and a minimum of 5. What are the amplitude and the midline?
Answer: B
Midline D = (17 + 5)/2 = 11 and amplitude = (17 - 5)/2 = 6. Choice A uses the full distance from minimum to maximum as the amplitude, which is twice the amplitude. Choice D uses the maximum as the amplitude.
Question 6 of 20 · Multiple Choice
Which function has amplitude 2, midline y = 5 and period π?
Answer: C
Period π requires B = 2π/π = 2, so y = 2 sin(2x) + 5. Choice A swaps the amplitude and the midline. Choice B has period 4π. Choice D has period 2π/π = 2, from putting the period itself in for B.
Question 7 of 20 · Multiple Choice
A situation starts at its minimum value at t = 0. Which form fits, with A > 0?
Answer: A
At t = 0, -A cos(0) + D = D - A, the minimum. Choice B starts at the maximum D + A. Choices C and D start on the midline, rising and falling.
Question 8 of 20 · Multiple Choice
A mass on a spring completes 2.5 oscillations per second. In a model y = A cos(Bt) + D with t in seconds, what is B?
Answer: C
B = 2π × frequency = 2π(2.5) = 5π. Choice A uses the frequency as B, which would give only 2.5/(2π), about 0.40 oscillation per second. Choice B is 2π divided by 2.5, the formula for B when 2.5 is the period, not the frequency. Choice D multiplies the frequency by π instead of 2π.
Question 9 of 20 · Multiple Choice
A Ferris wheel has a diameter of 60 ft, its lowest point is 4 ft above the ground, and it turns once every 6 minutes. A rider boards at the bottom. Which model gives the rider's height in feet after t minutes?
Answer: D
Midline 4 + 30 = 34 ft, amplitude 30 ft, B = 2π/6 = π/3, and the rider starts at the minimum, so h = 34 - 30 cos(πt/3). Check: h(0) = 4 and h(3) = 64. Choice A starts at the top. Choice C has period 12 minutes, from using π/6 instead of 2π/6.
Question 10 of 20 · Multiple Choice
A wave has a period of 0.125 second. What is its frequency?
Answer: B
Frequency is the reciprocal of the period: 1/0.125 = 8 cycles per second. Choice A confuses the period with the frequency. Choice C is B = 2π/0.125 = 16π, which is measured in radians per second, not cycles. Choice D multiplies the period by 2π instead of taking its reciprocal.
Question 11 of 20 · Multiple Choice
Which function has twice the frequency of y = sin(2πt)?
Answer: D
Doubling the frequency doubles B: from 2π to 4π, so the period halves from 1 to 1/2. Choice A doubles the amplitude. Choice B halves the frequency. Choice C moves the midline up 2.
Question 12 of 20 · Multiple Choice
In a model T(t) = A cos(Bt) + D of the average daily temperature through a year, which parameter equals the average of the highest and the lowest temperature?
Answer: C
D is the midline, the value halfway between the maximum and the minimum. Choice A, the amplitude, is half the difference between the highest and the lowest temperature. Choice B controls how many cycles occur per unit of time.
Question 13 of 20 · Multiple Choice
A speaker plays a tone of 330 hertz with pressure amplitude 0.8 Pa, midline 0, starting at 0 and rising. Which model fits, with t in seconds?
Answer: A
B = 2π(330) = 660π, and starting at the midline and rising calls for sine: p = 0.8 sin(660πt). Choice B uses the frequency as B. Choice C swaps the amplitude and the frequency. Choice D starts at the maximum.
Question 14 of 20 · Multiple Choice
A periodic graph has a maximum at (0, 9), the next maximum at (5, 9), and a minimum value of 1. Which function fits?
Answer: B
Midline (9 + 1)/2 = 5, amplitude (9 - 1)/2 = 4, period 5 so B = 2π/5, and a maximum at x = 0 calls for cosine. Choice A uses the full height 8 as the amplitude and the minimum as the midline. Choice C uses the period as B. Choice D starts on the midline, not at the maximum.
Question 15 of 20 · Short Answer
Write a function with amplitude 7, midline y = -2 and frequency 1/3 cycle per unit that starts on its midline and rises.
B = 2π(1/3) = 2π/3, and a start on the midline rising calls for sine: y = 7 sin(2πx/3) - 2. Check: y(0) = -2, and after a quarter period (x = 3/4) y = 7 sin(π/2) - 2 = 5, the maximum.
Question 16 of 20 · Short Answer
A bicycle wheel has radius 0.34 m. The valve stem is 0.30 m from the center of the wheel. The wheel turns 2 times per second, and the valve starts at its lowest point. Write the height h of the valve above the ground, in meters, after t seconds.
The center is 0.34 m above the ground, so the midline is h = 0.34. The amplitude is 0.30 m, B = 2π(2) = 4π, and a start at the minimum calls for -cos: h = -0.30 cos(4πt) + 0.34. Check: h(0) = 0.04 m, and at t = 0.25 s the valve is at the top, 0.64 m.
Question 17 of 20 · Short Answer
In a city, the longest day of the year has 14.8 hours of daylight and the shortest has 9.5 hours. Using t = 0 on the longest day and a period of 365 days, write a model for the hours of daylight D(t).
Midline (14.8 + 9.5)/2 = 12.15 hours, amplitude (14.8 - 9.5)/2 = 2.65 hours, B = 2π/365, and the model starts at its maximum: D(t) = 2.65 cos(2πt/365) + 12.15. Check: D(0) = 14.8 and D(182.5) = 9.5.
Question 18 of 20 · Short Answer
The water level in a reservoir, in meters, is modeled by L(t) = 1.2 sin(πt/6) + 3.5, where t is in months. State the amplitude, midline, period and frequency, and interpret each in the context.
Amplitude 1.2 m: the level rises and falls 1.2 m from its average. Midline L = 3.5 m: the average level. Period 2π/(π/6) = 12 months: the pattern repeats each year. Frequency 1/12 cycle per month. The level varies between 2.3 m and 4.7 m.
Question 19 of 20 · Short Answer
In y = A sin(Bx) + D, the number A can be negative. Explain why the amplitude cannot be negative, and what a negative A does to the graph.
The amplitude is a distance, from the midline to a maximum, so it is |A| and never negative. A negative A reflects the graph over the midline: y = -3 sin x + 1 starts on the midline and falls, while y = 3 sin x + 1 starts on the midline and rises. Both have amplitude 3.
Question 20 of 20 · Short Answer
A person's blood pressure varies between about 120 and 80 mmHg with a heart rate of 72 beats per minute. Model the pressure P after t minutes, starting at 100 mmHg and rising, and state what the frequency means.
Midline 100, amplitude 20, frequency 72 beats per minute, so B = 2π(72) = 144π: P(t) = 20 sin(144πt) + 100. The frequency means 72 full cycles (heartbeats) each minute; the period is 1/72 minute, about 0.83 second. A real pressure curve is not a perfect sine wave, so the model captures the range and the rate, not the exact shape.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.TF.B.5 mean?
HSF.TF.B.5 means choosing a sine or cosine function to model something that repeats, when the amplitude, frequency and midline are given or can be found from the situation. For a Ferris wheel 50 m across with its center 28 m high that turns every 12 minutes, students write h(t) = 28 - 25 cos(πt/6) and explain what 28, 25 and π/6 mean.
Is HSF.TF.B.5 Algebra 2 or Precalculus?
It is usually taught in Algebra II, and revisited in Precalculus. Unlike several other trigonometric function standards, it is not marked (+), so Common Core expects it of all students, not only those in advanced courses. It is also a modeling standard, which is why the lesson uses real contexts.
What is the difference between frequency and period?
The period is the time for one cycle, and the frequency is the number of cycles per unit of time, so each is the reciprocal of the other. A tide with a period of 12.4 hours has a frequency of 1/12.4 cycle per hour. A 440-hertz note has a frequency of 440 cycles per second and a period of 1/440 second.
Is B the frequency in y = A sin(Bx) + D?
No, B is 2π times the frequency. B is sometimes called the angular frequency and is measured in radians per unit of time. The frequency in cycles per unit is B/(2π), and the period is 2π/B. Some textbooks use the word "frequency" loosely for B, so it helps to say which one you mean in class and on assessments.
Should I use sine or cosine to model a situation?
Use whichever matches the situation at t = 0. If the quantity starts at its maximum, use A cos(Bt) + D; at its minimum, -A cos(Bt) + D; on the midline and rising, A sin(Bt) + D; on the midline and falling, -A sin(Bt) + D. Both functions have the same shape, so either can model any sinusoidal situation once a horizontal shift is allowed.
How do you find the amplitude and midline from a maximum and a minimum?
Midline = (max + min)/2 and amplitude = (max - min)/2. For a harbor where high tide is 3.1 m and low tide is 0.5 m, the midline is 1.8 m and the amplitude is 1.3 m. A frequent error is using the maximum, or the full distance from minimum to maximum, as the amplitude.
Does HSF.TF.B.5 include phase shift?
Not directly. The standard names amplitude, frequency and midline. Choosing between sine and cosine, and the sign of A, handles situations that start at a maximum, a minimum or the midline. A horizontal shift is needed when t = 0 falls between those points; many courses add it as an extension, together with graph transformations (HSF.BF.B.3).
What real-world examples fit this standard?
Good examples repeat at a steady rate: Ferris wheels and other rotating objects, a pendulum or a mass on a spring for a short time, sound waves, alternating current, tides and hours of daylight over a year. Temperature through a year and blood pressure are only roughly sinusoidal, which makes them good for discussing what a model leaves out.
What mistakes do students make when writing trigonometric models?
Common ones are using the frequency or the period as B, using the maximum as the amplitude, choosing sine when the situation starts at a maximum, mixing time units (a frequency per minute with t in seconds), and working in degree mode. Asking students to check the model at t = 0 and at a quarter period catches most of these.
How does HSF.TF.B.5 connect to other standards?
It builds on the unit circle and the periodicity of sine and cosine (HSF.TF.A.2 and HSF.TF.A.4), and on graphing trigonometric functions with their period, midline and amplitude (HSF.IF.C.7). The next step is HSF.TF.B.7, where students use inverse functions to answer questions such as "when is the rider 40 m high?" from the same models.
07
Related Standards
5 standards
These standards connect to HSF.TF.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSF.TF.A.2Prerequisite
Use the unit circle to extend trigonometric functions to all real numbers
