HSF.IF.B.4: Interpreting Key Features of Graphs and Tables in Context
In plain English: HSF.IF.B.4 is the Common Core functions standard that asks students to explain, in terms of the quantities, the key features of a graph or table that models a situation: intercepts, increasing, decreasing, positive and negative intervals, relative maximums and minimums, symmetry, end behavior and periodicity. Students also sketch graphs from verbal descriptions. It is usually taught in Algebra I and Algebra II.
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Interpret functions that arise in applications in terms of the context Also written as HSF-IF.B.4 or F-IF.4 · Official standard
Students learn to read a graph or a table of a function that models a real situation and to translate each key feature into a sentence about the quantities. A t-intercept is not just "where the curve hits the axis": it is the moment a ball lands. A relative maximum is the highest point a rider reaches, and a period is the time one full turn of a wheel takes. The lesson covers every feature the standard lists: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Students then work in the other direction: given only a verbal description of a relationship, they sketch a graph whose features match the story, label the axes with quantities and units, and mark each feature. Tables get special attention, because a table only shows some inputs, so features such as a zero or a maximum can often only be located between two table values.
Learning Objectives
By the end of this lesson, students will be able to:
Identify intercepts, relative maximums and minimums, and intervals of increase, decrease, positive and negative values on a graph, and state what each means in terms of the quantities
Estimate the same key features from a table of values and explain what the table can and cannot show
Describe symmetry, end behavior and periodicity of a model and interpret them in context
Sketch a graph from a verbal description, with labeled axes, units and marked key features
Prior Knowledge Required
Students should already be comfortable with:
Describing a graph qualitatively as increasing, decreasing, linear or nonlinear 8.F.B.5
Function notation and evaluating a function at an input HSF.IF.A.2
The meaning of domain and range HSF.IF.A.1
Plotting points and reading coordinates on a graph with scaled axes
Sketch on the board a phone battery graph for one day, with time on the horizontal axis and percent charge on the vertical axis, but do not give any numbers. Pose the prompt:
Warm-Up Prompt
"The graph shows your phone battery from 7 a.m. to 11 p.m. Where on the graph would you see the moment you plugged the phone in? What would it mean if the graph touched the horizontal axis? What would a flat piece of the graph tell you?"
Collect answers. Students usually say "the graph goes up when charging": name that an interval where the function is increasing. Touching the horizontal axis means the battery reached 0 percent, a horizontal intercept. A flat piece means the charge did not change, perhaps because the phone was off. Tell students that today they will name these features precisely and always say what they mean in the situation.
Direct Instruction20-25 minutes
Introduce the vocabulary with one model at a time, and insist on a context sentence for every feature: "The feature is ___, which means ___ (with units)."
Intercepts: the vertical intercept is the output when the input is 0 (a starting value). A horizontal intercept is an input where the output is 0.
Increasing, decreasing, positive, negative: give each as an interval of the input. Increasing means the output goes up as the input goes up. Positive means the output is above 0, so the graph is above the horizontal axis.
Relative maximum or minimum: an output that is greater (or less) than every nearby output. State both coordinates: when it happens and how large it is.
Symmetry: a vertical line where the graph is a mirror image, so inputs the same distance from that line have equal outputs.
End behavior: what the output does as the input grows very large (or very negative), for example "approaches 70°F".
Periodicity: the graph repeats the same pattern every fixed input length, called the period.
Graph: intercepts, maximum, symmetry
A ball is tossed from a rooftop. Its height in feet after t seconds is h(t) = -16t² + 48t + 64 (Diagram 1). Interpret the key features.
Equation: h(0) = 64: released 64 ft up. Max (1.5, 100): 100 ft after 1.5 s. h(4) = 0: lands at 4 s. Increasing on 0 < t < 1.5, decreasing on 1.5 < t < 4, symmetric about t = 1.5.
Table: positive, negative, relative extremes
Temperatures on a winter day in a mountain town: Table. Hours after midnight: 0, 3, 6, 9, 12, 15, 18, 21, 24. Temperature (°C): -4, -6, -7, -2, 3, 5, 1, -2, -5.
Equation: Negative from 0 h until between 9 and 12 h, positive until between 18 and 21 h, then negative. Lowest table value -7°C at 6 a.m.; highest 5°C at 3 p.m. Increasing from about 6 to 15 h.
End behavior
A cup of coffee cools in a 70°F room. Its temperature after t minutes is T(t) = 70 + 110(0.9)t. Interpret T(0) and the end behavior.
Equation: T(0) = 180: the coffee starts at 180°F. T(10) ≈ 108°F. T is decreasing, and as t grows, T(t) approaches 70°F: the coffee cools toward room temperature but never below it.
Sketch from a verbal description: periodicity
A Ferris wheel 40 m across has its center 22 m above the ground and turns once every 10 minutes. A rider boards at the lowest point. Sketch the rider's height for 20 minutes (Diagram 2).
Equation: Minimum 2 m at t = 0, 10, 20; maximum 42 m at t = 5 and 15; period 10 minutes; each turn is symmetric about its maximum.
For the table example, stop and ask: "Is -7°C really the coldest temperature of the day?" The table does not say. The true minimum could be between two listed hours, so students should write "the lowest value in the table" or "a relative minimum near 6 a.m.". The same is true for the zeros: the temperature crosses 0°C somewhere between 9 a.m. and noon.
Guided Practice15 minutes
Pairs analyze a table for a student-run coffee cart. The cart's weekly profit depends on the price of a cup:
Ask pairs to answer, with context sentences: Where is the profit negative, and what does that mean? (At $1.00 and at $4.00 the cart loses $40 a week. The break-even prices, where the profit is 0, lie between $1.00 and $1.50 and between $3.50 and $4.00, because the table changes sign there.) What is the relative maximum? (A profit of $95 at a price of $2.50, the best price in the table.) What symmetry do you see? (Prices the same distance from $2.50 give the same profit, for example $2.00 and $3.00 both give $80.) Circulate and listen for students who give a feature without units or who read the vertical intercept from a table that does not include p = 0.
Independent Practice15-20 minutes
Students sketch two graphs from verbal descriptions, label the axes with quantities and units, and mark every key feature. (1) Tide height at a pier: high tide of 6.2 ft at 3:00 a.m., low tide of 0.4 ft about 6.2 hours later, repeating about every 12.4 hours. Students mark the period, each high and low tide, and when the water is rising. (2) Spectators in a stadium: empty at 4 p.m., rising to 40,000 by the 7 p.m. kickoff, about level during the game, then falling to 0 by 11 p.m. Students mark the intercepts, the flat interval and the decreasing interval. Pairs then swap sketches and check each other's labels against the descriptions.
Closure5-10 minutes
Exit ticket: (1) W(t) gives a puppy's weight in kilograms t weeks after birth, and the graph has vertical intercept (0, 0.4). Interpret this point. (Answer: the puppy weighed 0.4 kg at birth.) (2) Sketch the number of people on a beach from 6 a.m. to 10 p.m. if the beach is empty at 6 a.m., is busiest at 2 p.m., and is nearly empty again by 10 p.m. Label the maximum and the intervals of increase and decrease. (3) In one sentence, describe the end behavior of the coffee model T(t) from class.
Differentiation Strategies
For Struggling Students
Give a feature-sentence frame card: "On the interval ___, the (quantity) is (increasing/decreasing), which means ___"
Start with graphs that show only one or two features, then add features one at a time
Color-code: one color for increasing intervals, another for decreasing, and shade above and below the axis for positive and negative
For Advanced Students
Give a table with uneven input spacing and ask students to explain which features they can state with certainty and which they can only estimate
Ask students to write a verbal description whose graph has exactly two relative maximums, one relative minimum and end behavior that levels off, then trade with a partner to sketch
Ask students to find a real data set online (daily high temperatures, tide tables) and describe its periodicity and symmetry
Assessment Guidance
What to Look For
Listen for context and units in every answer: "a maximum of 100 feet at 1.5 seconds", not "the top is at (1.5, 100)". Check that intervals are given in terms of the input (time, price), not the output. When students work from a table, look for honest language such as "between 9 and 12 hours" or "the lowest value in the table". In sketches, check that axes are labeled with quantities and units and that each feature named in the description appears on the graph.
02
Classroom Activities
3 Activities
1
Graph, Table, Story Card Sort
20 minGroups of 3
Each group receives 18 cards: 6 verbal descriptions, 6 tables and 6 unlabeled graphs of the same six situations. Groups match each description to its table and graph, then label every key feature on the graph card in context.
The Six Situations
An oven preheating to 350°F. Table (minutes, °F): (0, 70), (4, 210), (8, 300), (12, 340), (16, 350). Feature: increasing, levels off near 350°F
The seat of a swing kept going at a steady height by a pusher. Table (seconds, meters): (0, 1.5), (0.8, 0.5), (1.6, 1.5), (2.4, 0.5), (3.2, 1.5). Feature: periodic, period 1.6 s
A walk to a friend's house 1.2 km away and straight back at the same pace. Table (minutes, km from home): (0, 0), (10, 0.6), (20, 1.2), (30, 0.6), (40, 0). Features: maximum, symmetry about 20 minutes
The value of a car. Table (years, dollars, rounded to $10): (0, 28,000), (2, 19,600), (4, 13,720), (6, 9,600), (8, 6,720). Feature: decreasing, approaches 0
A reservoir over one year. Table (month, percent full): (Jan, 60), (Apr, 85), (Jul, 70), (Oct, 45), (next Jan, 58). Features: relative maximum in spring, relative minimum in fall
A diver's height relative to the water surface. Table (seconds, meters): (0, 3), (1, 0), (2, -2.5), (3, -1.2), (4, 0). Features: zeros, negative interval, minimum
Procedure
Deal the description cards face up and place the tables and graphs in a pile
Groups match all 18 cards into six sets and write, on the back of each graph card, two key features with a context sentence for each
Each group explains to another group one set where the table alone was not enough to decide the match
Discussion Questions
Which two situations have symmetric graphs, and why does the story produce the symmetry?
The car table never reaches 0. Could the car's value ever be negative? What does the end behavior tell you?
How can you find the swing's period from the table alone?
2
Describe, Draw, Compare
20 minPairs
Partner A holds a graph that Partner B cannot see and describes it using only key-feature vocabulary and the context. Partner B sketches from the description. The pair then compares the two graphs and finds which words were missing or unclear.
Graph Cards for Partner A
Round 1: the water level in a rain barrel over a week with two storms (two increasing intervals, a relative maximum after each storm, decreasing as water is used)
Round 2: the number of cars on a highway over 48 hours (two rush-hour maximums per day, pattern repeating every 24 hours)
Round 3: a bank balance that starts at $50, becomes negative after a large purchase, then rises above 0 after a paycheck
Procedure
Partner A may name features (intercepts, intervals, extremes, symmetry, end behavior, period) and quantities, but may not use gestures or say "it looks like"
Partner B labels axes with quantities and units before sketching
Partners compare and write one feature that matched and one that did not; roles switch each round
Modification for Distance Learning
Partner A shares the description in the chat; Partner B sketches on a shared whiteboard. The class votes on which sketch best matches each description.
3
Table Detectives: Hours of Daylight
20-25 minGroups of 3-4
Groups analyze a table of approximate daylight hours on the 21st of each month for a city at about 47° north latitude (values rounded, for classroom use) and describe the key features that the table shows.
Find the relative maximum and minimum in the table and say what each means (longest and shortest days, near the June and December solstices)
Describe the intervals where daylight increases and decreases
Explain why the function is periodic with a period of about 12 months, and predict the daylight for next January
Look for approximate symmetry: which months have nearly equal daylight, and around which month are they balanced?
Challenge Variation
Give groups the same table for a city near the equator (every month close to 12 hours) and ask how the features change. Groups sketch both graphs on one set of axes.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Key Features of a Ball's Height Graph
Graph of h(t) = -16t² + 48t + 64 for 0 ≤ t ≤ 4, drawn to scale. Each marked point is a key feature with a meaning in context: the release height, the highest point and the landing time. The dashed line t = 1.5 is the line of symmetry.
Diagram 2: A Sketch from a Verbal Description, Ferris Wheel Height
Height of a rider who boards at the bottom of a 40 m Ferris wheel whose center is 22 m high and which turns once every 10 minutes, drawn to scale. The graph is periodic with period 10 minutes, with a minimum of 2 m and a maximum of 42 m in every turn.
04
Homework Assignment
~30 min
HSF.IF.B.4 Homework: Key Features in Context
Directions: Answer in complete sentences that name the quantities and units. You may use a graphing calculator or app to graph the functions in Parts 1 and 2. For Part 3, label both axes with quantities and units and mark every key feature on your sketch.
Part 1: Graphs and Tables (Problems 1-2)
The height in meters of a model rocket t seconds after launch is h(t) = -4.9t² + 29.4t. Graph the function for 0 ≤ t ≤ 6. Find and interpret both t-intercepts and the maximum, give the intervals where the height is increasing and decreasing, and describe the symmetry of the graph.
A new food truck records its monthly profit, in thousands of dollars: Table. Month: 1, 2, 3, 4, 5, 6, 7, 8. Profit: -12, -7, -2, 3, 6, 7, 5, 1. (a) In which months was the profit negative, and between which months did the profit change from negative to positive? (b) Where does the table show a relative maximum, and what does it mean? (c) On which interval is the profit increasing?
Part 2: End Behavior and Periodicity (Problems 3-4)
The concentration of a medicine in a patient's blood t hours after a dose is C(t) = 20t/(t² + 4) milligrams per liter. Make a table for t = 0, 1, 2, 4, 8 and 20. Find the maximum concentration and when it occurs, and describe the end behavior in context.
The depth of water in a harbor t hours after midnight is d(t) = 12 + 5sin(πt/6) feet. Use a graph for 0 ≤ t ≤ 24. What is the period, and what does it mean? Give the maximum and minimum depths and the first time each occurs. Over which intervals between t = 0 and t = 12 is the water rising?
Part 3: Sketching from a Description (Problems 5-6)
A kayaker paddles away from a dock at a steady pace for 40 minutes until she is 3 km away, stops for 20 minutes for lunch, then paddles straight back at a faster steady pace, reaching the dock 30 minutes later. Sketch her distance from the dock over time. Label the intercepts, the maximum, and the intervals where the distance increases, stays constant and decreases.
An underwater survey drone starts at the surface of a lake. It descends at a steady rate to 60 m below the surface in 4 minutes, holds that depth for 8 minutes, then rises at the same rate back to the surface. Sketch its elevation relative to the surface (negative below the surface). Label the intercepts, the interval where the elevation is negative, the minimum, and the line of symmetry.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Identifying Features
All requested features found with correct values or intervals
Most features correct, one missing or misread
Features missing or incorrect
Interpreting in Context
Every feature explained with quantities and units
Some explanations lack units or context
No interpretation given
Sketching
Axes labeled with units; all described features shown
Shape correct but labels or a feature missing
Sketch does not match the description
Reasoning with Tables
States clearly what the table shows and what is only estimated
Correct values but no comment on estimates
Misreads the table
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then open the explanation to check your reasoning. Your score updates as you go, and Reset quiz clears every answer so the quiz can be used 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
Question 1 of 20 · Multiple Choice
B(m) is the balance in dollars on a prepaid transit card after m rides. Its graph passes through (0, 40) and (16, 0). What does the point (16, 0) mean?
Answer: B
A horizontal intercept is an input where the output is 0: after m = 16 rides, B = 0 dollars. (Each ride costs 40/16 = $2.50.) Choice A reads the input as if it were the output. Choice D confuses this point with the vertical intercept (0, 40), which is the starting balance of $40.
Question 2 of 20 · Multiple Choice
The graph of a hiker's elevation E(t), in feet, t hours after she starts has a relative maximum at (2.5, 4800). Which statement interprets this feature?
Answer: C
A relative maximum gives an output that is greater than the outputs at nearby inputs: 2.5 hours in, she was at a high point of 4,800 feet, for example a summit. Choice A treats elevation as distance walked. Choice B confuses a height with a rate of change. Choice D swaps the coordinates.
Question 3 of 20 · Multiple Choice
The table shows the temperature at a mountain weather station. Table. Hours after 4 a.m.: 0, 2, 4, 6, 8, 10. Temperature (°C): -6, -1, 2, 3, 1, -4. Based on the table, over which interval is the temperature increasing?
Answer: A
The values rise from -6 to 3 as the input goes from 0 to 6 hours, then fall. So the temperature increases from 4 a.m. to about 10 a.m., as far as the table shows. Choice C is roughly where the temperature is positive, not where it is increasing. Choice D is the decreasing interval.
Question 4 of 20 · Multiple Choice
A cave diver's elevation relative to the water surface is shown in the table. Table. t (min): 0, 5, 10, 15, 20. Elevation (m): 0, -12, -18, -12, 0. What do the zeros at t = 0 and t = 20 mean?
Answer: A
An elevation of 0 m means the diver is at the surface, so the dive starts at the surface and ends there after 20 minutes. Choice C describes the minimum, which is -18 m at 10 minutes. Choice B confuses an output of 0 with a rate of change of 0.
Question 5 of 20 · Multiple Choice
Using the same cave diver table, which statement about symmetry is supported?
Answer: B
Inputs the same distance from t = 10 have equal outputs: at 5 and 15 minutes the elevation is -12 m, and at 0 and 20 minutes it is 0 m. So the table is symmetric about t = 10. Choice A describes a horizontal line of symmetry, which a function graph of this kind does not have. Choice D confuses symmetry with sign.
Question 6 of 20 · Multiple Choice
A(t) is the amount of caffeine, in milligrams, in a person's body t hours after a drink. As t increases without bound, A(t) approaches 0. What does this end behavior mean?
Answer: D
End behavior describes the output as the input gets very large: the amount left approaches 0 mg. Choice A is impossible in context and is not what "approaches 0" means. Choice C confuses approaching 0 with reaching 0 at a specific time.
Question 7 of 20 · Multiple Choice
The height of a point on a wind turbine blade is a periodic function of time with a period of 4 seconds. The point is at its highest at t = 1 second. When is it next at its highest?
Answer: C
A periodic function repeats every period, so the next maximum is at 1 + 4 = 5 seconds. Choice B adds half a period: at t = 3 the point is at its lowest. Choice D uses two periods starting from 0 instead of one period after t = 1.
Question 8 of 20 · Multiple Choice
A buoy bobs on waves. The table gives its height relative to the calm water level. Table. t (s): 0, 1, 2, 3, 4, 5, 6, 7, 8. Height (m): 0, 0.6, 0, -0.6, 0, 0.6, 0, -0.6, 0. What is the period of the motion?
Answer: C
The full pattern 0, 0.6, 0, -0.6 repeats every 4 seconds: the buoy is at 0.6 m at t = 1 and again at t = 5. Choice B is the time between consecutive zeros, which is only half a cycle, since the buoy is moving up at t = 0 and down at t = 2.
Question 9 of 20 · Multiple Choice
A skier rides a chairlift from the base to the top of a run at a steady speed, then skis straight down. Which set of features should a sketch of her elevation over time show?
Answer: B
Elevation increases steadily during the lift ride (a straight segment), reaches a relative maximum at the top, then decreases back to the base elevation. Choice A reverses the story. Choice C would mean the pattern repeats, but the description is one trip. Choice D confuses the start and end values with the whole graph.
Question 10 of 20 · Multiple Choice
A metal baking tray at 70°F is placed in an oven set to 425°F. Which feature should a sketch of the tray's temperature over time show?
Answer: A
The tray warms quickly at first, then more slowly as it nears the oven temperature, so the end behavior is "approaches 425°F". Choice B ignores that the tray cannot get hotter than the oven. Choice C describes cooling in a room, the reverse situation.
Question 11 of 20 · Multiple Choice
P(x) is a company's profit in thousands of dollars when it sells x hundred units. P(x) is negative for 0 ≤ x < 3 and positive for x > 3. What does this mean?
Answer: B
x is measured in hundreds of units, so x = 3 means 300 units, the break-even point. Negative output means a loss. Choice A ignores the unit "hundred". Choice C confuses negative with decreasing: a loss can shrink while still being a loss.
Question 12 of 20 · Multiple Choice
The table shows the number of students in a school library each hour. Table. Time: 8 a.m., 9, 10, 11, 12 p.m., 1, 2, 3. Students: 40, 25, 18, 30, 72, 55, 20, 64. At which times does the table show relative minimums?
Answer: C
A relative minimum is lower than its neighbors: 18 at 10 a.m. (between 25 and 30) and 20 at 2 p.m. (between 55 and 64). Choice A gives only the lowest value overall and misses the second relative minimum. Choice B is the relative maximum of 72. Choice D lists the endpoints of the table.
Question 13 of 20 · Multiple Choice
A suspension bridge cable's height y, in meters, at horizontal distance x from the left tower is symmetric about x = 210. The cable is 58 m high at x = 90. What is its height at x = 330?
Answer: D
x = 90 and x = 330 are both 120 m from x = 210 (210 - 120 = 90 and 210 + 120 = 330), so symmetry gives the same height, 58 m. Choice C ignores what symmetry guarantees. Choices A and B halve or double the height without a reason.
Question 14 of 20 · Multiple Choice
S(d) is the depth of snow in centimeters at a ski resort d days after March 1. Its graph has vertical intercept (0, 145). What does this mean?
Answer: D
The vertical intercept is the output when the input is 0, and d = 0 is March 1, so S(0) = 145 means 145 cm of snow on March 1. Choice A reads the output as an input. Choice B confuses a value with a rate.
Question 15 of 20 · Short Answer
At a bike-share dock, the table shows the number of bikes compared with 6 a.m. (bikes returned minus bikes taken). Table. Time: 6 a.m., 7, 8, 9, 10, 11. Change in bikes: 0, -5, -11, -8, -2, 3. Give the interval where the change is negative, find the relative minimum, and interpret both.
The change is negative from 7 a.m. until some time between 10 and 11 a.m.: during that time the dock has fewer bikes than at 6 a.m., because commuters take bikes. The table changes sign between 10 a.m. (-2) and 11 a.m. (3), so that is where the zero is. The relative minimum is -11 at 8 a.m.: the dock had 11 fewer bikes than at 6 a.m., the lowest value in the table.
Question 16 of 20 · Short Answer
Sketch the number of cars in a school parking lot from 6 a.m. to 6 p.m.: empty at 6 a.m., filling to 280 cars by 8 a.m., staying about level until 2:30 p.m., dropping to about 20 cars by 4 p.m., then to 0 by 6 p.m. List the key features your sketch should show.
The sketch starts at (6 a.m., 0), increases to 280 cars at 8 a.m., is about constant at 280 from 8 a.m. to 2:30 p.m. (the maximum), decreases steeply to 20 at 4 p.m., then decreases slowly to 0 at 6 p.m. Axes: time of day and number of cars. The graph is never negative, because a count of cars cannot be below 0.
Question 17 of 20 · Short Answer
A new employee's typing speed after w weeks of practice is S(w) = 70 - 40(0.8)w words per minute. Find and interpret S(0), say whether S is increasing or decreasing, and describe the end behavior.
S(0) = 70 - 40 = 30 words per minute, the speed at the start. Because (0.8)w gets smaller as w grows, S is increasing. As w increases without bound, (0.8)w approaches 0, so S(w) approaches 70 words per minute: the employee keeps improving, but more slowly, toward about 70 words per minute.
Question 18 of 20 · Short Answer
A bicycle pedal moves between 10 cm and 38 cm above the ground, making one full turn every 0.8 seconds at a steady cadence. The pedal is highest at t = 0.2 s. Give the period, the first three times after t = 0 when the pedal is highest, and the number of turns in one minute.
The period is 0.8 seconds, one turn of the pedal. The pedal is highest at t = 0.2, 1.0 and 1.8 seconds (add 0.8 each time), always at 38 cm. In 60 seconds it makes 60/0.8 = 75 turns.
Question 19 of 20 · Short Answer
The height of a water jet from a fountain is h(x) = -0.5(x - 3)² + 4.5 meters, where x is the horizontal distance in meters from the nozzle. Find the x-intercepts and the maximum, and use symmetry to find h(5) from h(1).
h(x) = 0 when (x - 3)² = 9, so x = 0 and x = 6: the water leaves the nozzle at ground level and lands 6 m away. The maximum is 4.5 m at x = 3. The graph is symmetric about x = 3, and 1 and 5 are both 2 m from 3, so h(5) = h(1) = -0.5(4) + 4.5 = 2.5 m.
Question 20 of 20 · Short Answer
A wildlife biologist describes a deer population: 200 deer when a reserve opens, growing quickly at first, then more slowly, and leveling off near 800 deer, the most the reserve can support. Sketch the population over 20 years and name the end behavior and the vertical intercept.
The sketch starts at the vertical intercept (0, 200) and is increasing the whole time, steep at first and then flatter. The end behavior: as time increases, the population approaches 800 deer from below, so the graph levels off near a horizontal line at 800. There is no maximum inside the interval, and the graph is never periodic.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.IF.B.4 mean?
HSF.IF.B.4 means students can look at a graph or a table of a function that models a real situation and explain each key feature in terms of the quantities. The features named in the standard are intercepts, intervals of increase, decrease, positive and negative values, relative maximums and minimums, symmetries, end behavior and periodicity. Students must also go the other way and sketch a graph from a verbal description.
Is HSF.IF.B.4 Algebra 1 or Algebra 2?
Both. Many course sequences introduce it in Algebra I with linear, quadratic and exponential models, and return to it in Algebra II with polynomial, rational and trigonometric models, where end behavior and periodicity become more important. The standard itself does not name a function type, so any model the course uses can be interpreted with it.
What does "interpret in terms of the quantities" mean?
It means every feature is explained with the context and units, not only with coordinates. "The maximum is (1.5, 100)" is a description of the graph. "The ball reaches its greatest height, 100 feet, 1.5 seconds after it is released" is an interpretation. Teachers should ask for the second kind of sentence every time.
What is the difference between a relative maximum and a maximum?
A relative maximum is higher than the points near it; the overall (absolute) maximum is the highest point on the whole domain. A graph of highway traffic can have two relative maximums each day, the morning and evening rush hours, and only the larger of the two is the day's overall maximum.
How can students find key features from a table instead of a graph?
Look for changes: where outputs switch from going up to going down (a relative maximum), where they change sign (a zero lies between those inputs), where values repeat in a pattern (periodicity), and where equal outputs sit at equal distances from one input (symmetry). Students should say "between" or "in the table" when the exact value is not listed, because a table only shows some inputs.
What are common mistakes with increasing and decreasing intervals?
A common error is giving the interval in terms of the output, such as "increasing from 64 to 100", instead of the input, "increasing from 0 to 1.5 seconds". Another is confusing positive with increasing: a company's profit can be positive while it is going down. Students also mix up a function that is negative with one that is decreasing.
What does end behavior mean in a real situation?
End behavior tells what happens to the quantity in the long run, as the input keeps growing. For a cooling drink, the temperature approaches room temperature. For a car's value, it approaches $0. For a population with limited space, it levels off at the most the habitat can support. Many real models only make sense on a limited domain, so students should also say where the model stops applying.
Which real situations are periodic?
Anything that repeats on a regular cycle: a Ferris wheel rider's height, tides, hours of daylight over a year, a pedal on a bicycle, or a point on a fan blade. The period is the input length of one full cycle, for example 10 minutes per turn or about 12.4 hours between high tides. The trigonometric models behind these situations come later, in HSF.TF.B.5.
How should students sketch a graph from a verbal description?
Start by naming the two quantities and choosing which is the input. Label both axes with units, then mark the points the story gives (starting value, highest point, when it ends) before drawing. Then connect the points with the right shape: steady change is a straight segment, "quickly at first, then more slowly" is a curve that flattens, and "repeats" means a periodic pattern. The sketch does not need an equation.
How is HSF.IF.B.4 tested?
Typical items give a graph or a table in a context and ask what a point, an interval or a feature means, or ask students to choose or draw a graph that matches a description. Answers usually need the context and units, not only a number. Questions that ask students to interpret graphs of linear and nonlinear models in context appear in the Algebra and Advanced Math domains of the digital SAT.
07
Related Standards
6 standards
These standards connect to HSF.IF.B.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.B.5Prerequisite
Describe a functional relationship qualitatively from a graph and sketch one from words
