8.F.B.5: Describing and Sketching Graphs of Functions
In plain English: 8.F.B.5 is the Common Core grade 8 math standard that asks students to describe a function (a rule that gives each input exactly one output) by reading the shape of its graph: where it is increasing or decreasing, and where it is linear or nonlinear. Students also sketch a graph that shows the features of a situation described in words. It is part of the functions unit of Grade 8 Math.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Common Core State Standards for Mathematics · Domain: Functions (F) · Cluster: Use functions to model relationships between quantities. Also written as 8.F.5 · Official standard
Students learn to read the story a graph tells without doing exact calculations, and to draw a graph that tells a story they are given. This kind of description is called qualitative: it talks about the shape and behavior of a graph in words (going up, going down, flat, straight, curved) rather than with exact numbers. The key words are increasing (as x goes right, y goes up), decreasing (as x goes right, y goes down), constant (y stays the same), linear (the graph is a straight line, so y changes by the same amount for each equal step in x) and nonlinear (the graph curves, so the change in y is not the same for equal steps).
Students describe graphs piece by piece, using intervals (stretches of x-values, such as "from 8 to 14 minutes"), and compare how steep the pieces are. Then they turn stories about walking, filling, cooling and saving into sketches that show every feature the story names. A sketch needs labeled axes and the right shape, not an exact scale.
Learning Objectives
By the end of this lesson, students will be able to:
Name the intervals where a function is increasing, decreasing or constant by reading its graph
Decide from a graph whether a function is linear or nonlinear, and explain how they know
Compare the steepness of pieces of a graph and say what it means in the situation
Sketch a graph with labeled axes that shows all the features of a situation described in words
Prior Knowledge Required
Students should already be comfortable with:
Knowing that a function gives each input exactly one output 8.F.A.1
Knowing that a linear function y = mx + b has a straight-line graph 8.F.A.3
Plotting and reading ordered pairs on a coordinate grid 6.NS.C.8
Describing how one quantity depends on another, such as distance and time 6.EE.C.9
Give each student a small piece of grid paper. Draw two axes on the board: time across the bottom, "height above the ground" up the side. No numbers.
Warm-Up Prompt
"A child climbs the ladder of a playground slide, sits at the top for a moment, then slides down to the ground. Sketch the child's height above the ground against time. Which part of your graph is the steepest? Why?"
Share three sketches. A good sketch goes up (the climb), stays flat (sitting at the top), then goes down more steeply than it went up (the slide is faster than the climb). Some students draw the shape of the slide itself. Use that picture to make the main point of the lesson: a graph shows how one quantity changes as another changes. It is not a picture of the place.
Direct Instruction20 minutes
Build an anchor chart with a tiny sketch next to each word:
Qualitative description: telling how a graph behaves in words, without exact values: "the distance goes up steadily, then stays the same."
Interval: a stretch of x-values, such as "from x = 8 to x = 14." Describe a graph one interval at a time, always reading from left to right.
Increasing, decreasing, constant: on an interval, the function is increasing if y goes up as x goes right, decreasing if y goes down, and constant if the graph is flat.
Linear or nonlinear: a piece is linear if it is a straight line, so equal steps in x give equal changes in y (a steady rate). It is nonlinear if it curves, so the changes are not equal.
Steepness: on the same axes, a steeper piece means y changes faster. Use the rate of change (change in y ÷ change in x) to compare two straight pieces.
Work through the five examples, using Diagram 1 for the first and Diagram 2 for the second.
Describing a graph piece by piece (Diagram 1)
A graph of Rosa's distance from home goes in a straight line from (0, 0) to (8, 600), stays flat to (14, 600), goes in a straight line down to (18, 0), and stays at 0 until 20 minutes.
Equation: Increasing and linear from 0 to 8 minutes (she walks to the store at 600 ÷ 8 = 75 m per minute). Constant from 8 to 14 minutes (she is in the store). Decreasing and linear from 14 to 18 minutes, and steeper: 600 ÷ 4 = 150 m per minute, so she runs home. Constant at 0 from 18 to 20 minutes.
Linear or nonlinear? (Diagram 2)
Water flows at the same steady rate into a straight-sided glass and into a vase with straight, slanted sides that is 4 cm across at the bottom and 12 cm across at the top. Both fill to 16 cm in 40 seconds.
Equation: The glass graph is linear: the water rises 4 cm every 10 seconds. The vase graph is increasing but nonlinear: it rises about 7.7 cm in the first 10 seconds and only about 2.1 cm in the last 10 seconds, because the same amount of water spreads over a wider space near the top.
Sketching from a description
A phone shows 100% charge at 7 am. It loses charge at a steady rate during the school day, reaching 40% at 3 pm. It is then plugged in and charges quickly to 90% by 4 pm, and it stays at 90% while it is off until 6 pm. Sketch charge against time.
Equation: Label the axes "time" and "charge (%)." Sketch a straight piece going down from 7 am to 3 pm, a much steeper piece going up from 3 pm to 4 pm, and a flat piece from 4 pm to 6 pm. The key features are the order of the pieces and which piece is steepest.
A nonlinear graph that goes up, then down
A soccer ball is kicked high into the air and lands on the field. Describe the graph of its height against time.
Equation: Increasing, then decreasing, and nonlinear the whole time: the ball rises fast, slows near the top, then falls faster and faster. The highest point of the graph is when the ball is highest. The graph is a curved hill, not two straight pieces.
Matching a story to a curve
A cup of hot cocoa is left on a table in a room at 21 °C. Describe the graph of its temperature against time.
Equation: Decreasing and nonlinear. It drops fastest at the start, when the cocoa is much hotter than the room, then more and more slowly, and levels off close to 21 °C. It never drops below room temperature.
After Example 2, if you have a glass and a narrow-bottom vase, pour water into both at the same steady rate and let students watch the heights. Stress three habits: read left to right, name each interval before describing it, and use the words "linear" and "nonlinear" only after checking whether equal steps give equal changes.
Guided Practice15 minutes
Pairs work through four items. One partner describes or sketches, the other checks each feature against the definitions on the anchor chart, then they switch.
Guided practice items with answers
Item
Answer
A graph of the money in a lunch account (dollars) against weeks is flat from (0, 20) to (4, 20), then a straight line down to (9, 0). Describe it.
Constant from week 0 to week 4. Decreasing and linear from week 4 to week 9, at 20 ÷ 5 = $4 per week
A kettle heats water from 20 °C to 100 °C in 3 minutes, boils at 100 °C for 1 minute, and is then turned off and cools. Sketch temperature against time.
Increasing to 100 °C, a flat piece at 100 °C, then decreasing, fast at first and then slowly (curved)
A graph of the area of a square (y) against its side length (x) passes through (0, 0), (1, 1), (2, 4) and (3, 9). Is it linear?
No. For each step of 1 in x, y goes up 1, then 3, then 5. It is increasing and nonlinear
A skydiver jumps from a plane. Height drops faster and faster until the parachute opens, then drops slowly at a steady rate to the ground. Sketch height against time.
Decreasing the whole time: a curve that gets steeper, then a straight, less steep piece down to 0
Listen for students who describe the graph from right to left, and for students who call a curved graph "linear" because it only goes up.
Independent Practice15 minutes
Students work alone, then compare with a partner.
Independent practice items with answers
Item
Answer
A graph passes through (0, 3), (2, 7), (4, 11) and (6, 15), and nothing else changes between them. Describe it.
Increasing and linear: y goes up 4 for every 2 in x
A graph passes through (0, 64), (1, 32), (2, 16) and (3, 8) along a smooth curve. Describe it.
Decreasing and nonlinear: the drops are 32, 16, 8, so it falls more slowly each step
Sketch the number of cars in a school parking lot from 6 am to 6 pm on a school day.
Increasing quickly around 7-8 am, about constant during the day, decreasing quickly after school, then slowly
A candle burns at a steady rate for 2 hours and is then blown out. Sketch its height against time.
Decreasing and linear for 2 hours, then constant
Write a story that fits this graph: increasing and linear, then constant, then decreasing and linear but steeper than the first piece.
Answers vary, for example: a student saves money at a steady rate, keeps the savings for a while, then spends them faster
Closure5-10 minutes
Exit ticket: (1) A graph of the gas in a car's tank against time goes down in a straight line, then rises very steeply, then goes down in a straight line again. Tell the story. (The car uses gas at a steady rate, is filled up at a gas station, then drives on.) (2) A ball is dropped from your hand and bounces twice. Sketch its height above the floor against time. (Down and up, with the second bounce lower; every piece curved.) (3) Finish the sentence: "A graph is nonlinear when ..."
Differentiation Strategies
For Struggling Students
Give a strip of paper to slide from left to right across a graph, and have students say "up," "down" or "flat" as the edge moves
Provide a word bank (increasing, decreasing, constant, linear, nonlinear, steeper, faster, slower) and a sentence frame: "From ___ to ___, the graph is ___, which means ___"
Start sketches with the axes already labeled and the story cut into numbered parts, one part per piece of the graph
For Advanced Students
Ask for a sketch of the water height in a bottle whose shape changes three times (wide, narrow neck, wide again), and have a partner guess the bottle from the graph
Ask students to write two different stories for the same graph, one about distance and one about money
Extension (beyond this standard, see HSF.IF.B.6): on Diagram 2, have students compute the vase's average rise per second from 0 to 10 seconds and from 30 to 40 seconds, and explain what the difference means
Assessment Guidance
What to Look For
A strong description names every interval, uses one feature word for each (increasing, decreasing, constant) and says whether the piece is linear or nonlinear, with a reason. A strong sketch has labeled axes, the pieces in the right order and the right relative steepness. Watch for three errors: drawing a picture of the place instead of a graph, calling any graph that only goes up "linear," and drawing a vertical segment for a sudden change.
02
Classroom Activities
3 Activities
1
Graph and Story Card Match
15 minPairs
Each pair gets 12 cards: 6 graph cards (sketches with no numbers) and 6 story cards. Pairs match each graph with its story and write the features that decided the match on the back of the graph card.
The Graph Cards (what each sketch shows)
Graph A: Rises in a straight line, then stays flat
Graph B: Falls in a straight line all the way to 0
Graph C: Rises in a curve that gets steeper and steeper
Graph D: Flat, then falls in a straight line, then flat at 0
Graph E: Rises in a curve that gets less and less steep, then levels off
Graph F: Rises, then falls, then rises again, all in straight pieces
The Story Cards
Story 1: The height of a candle that burns at a steady rate until it is gone
Story 2: A savings account that earns more each year than the year before
Story 3: A bathtub that is filled at a steady rate, then the water is turned off
Story 4: The height of a pumpkin plant over one summer: fast growth at first, then slower, then almost none
Story 5: Water in a pool before the drain is opened, while it drains at a steady rate, and after it is empty
Story 6: A hiker's elevation (height above sea level): up a hill, down into a valley, then up the next hill, each at a steady pace
Answer Key
Graph A with Story 3
Graph B with Story 1
Graph C with Story 2
Graph D with Story 5
Graph E with Story 4
Graph F with Story 6
Discussion Questions
Graphs C and E both increase the whole time. Which feature tells them apart?
Which two graphs end at a value of 0? How do their stories show that?
Graph F goes up twice. Does that mean the hiker climbed the same hill twice?
Modification for Distance Learning
Put the sketches and stories on a shared slide and have pairs draw connecting arrows, then type one feature word beside each arrow.
2
Walk the Graph
25 minGroups of 4
Groups mark a straight line on the floor with masking tape every half meter, starting at a wall. One student walks, one times with a stopwatch, one calls out the walker's distance from the wall every 2 seconds, and one records. Each group then graphs the walk and describes it, piece by piece.
Sample Data (invented)
An invented group recorded the walker's distance from the wall at 0, 2, 4, 6, 8, 10 and 12 seconds: 1.0, 1.8, 2.6, 2.6, 2.6, 1.4 and 0.2 meters.
Procedure
Walk 1: the walker follows this story: "walk away from the wall slowly and steadily, stop and stand still, then walk back toward the wall faster"
Plot the (time, distance) pairs on grid paper and join them with straight pieces
Label each interval of the graph as increasing, decreasing or constant, and mark the steepest piece
Walk 2: the recorder draws a secret sketch; the walker must walk it, and the group compares the new graph with the sketch
Discussion Questions
In the sample, the walker moves away at 0.4 m per second and back at 0.6 m per second. Which piece of the graph is steeper, and does that match the story?
In the sample, the distance stays at 2.6 m from 4 to 8 seconds. What was the walker doing?
Could any walk make a graph with a vertical segment? Why or why not?
Challenge Variation
Ask the walker to make a curved graph: start slowly and speed up while walking away. Groups decide whether their recorded points show a nonlinear piece.
3
Sketch, Swap and Critique
15 minPairs
Each student sketches the graphs for two short stories, then swaps with a partner. The partner checks each sketch against a feature checklist and writes one question or fix. Pairs finish by critiquing a sample sketch that has mistakes.
The Stories
Story 1: The number of people at a school fair. Nobody is there before it opens at noon, people arrive quickly for the first hour, the number stays about the same all afternoon, and people leave quickly at 5 pm
Story 2: A student's heart rate in gym class. It is steady while the class sits for directions, rises quickly during a warm-up jog, stays about the same while the jog goes on, rises quickly again during a short sprint, and then drops, fast at first and then more slowly, while the class rests
Feature Checklist
Both axes are labeled with the quantity (and a unit where it helps)
The pieces appear in the same order as the story, from left to right
Each piece is increasing, decreasing or constant as the story says, and straight or curved as the story says
Faster changes are steeper; no piece is vertical
Sample Sketch to Critique
For Story 2, a student drew: a flat piece, a rise, a flat piece, a vertical segment for the sprint, and a straight line down to zero. Pairs should find three problems: the sprint needs a steep rise that takes some time, not a vertical segment; the drop at the end should curve (fast at first, then slower); and it should level off at a resting heart rate, not reach zero.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Describing a Graph One Interval at a Time
Rosa's distance from home, drawn to scale. Piece A rises 600 m in 8 minutes and piece C falls 600 m in 4 minutes, so C is twice as steep. Pieces B and D are flat, so the distance is constant there: Rosa is in the store during B and back home during D.
Diagram 2: Linear and Nonlinear Graphs of the Same Kind of Story
Water flows at the same steady rate into a straight-sided glass (dashed, linear) and into a vase that is narrow at the bottom (solid, nonlinear). Both graphs increase. Only the glass rises by the same amount, 4 cm, in every 10 seconds. The vase rises about 7.7 cm, 3.6 cm, 2.6 cm and 2.1 cm in the four 10-second intervals, so its curve gets less steep. The curve is computed from the vase shape: 4 cm across at the bottom, 12 cm across at the top and 16 cm tall.
04
Homework Assignment
~30 min
8.F.B.5 Homework: Reading and Sketching Graphs
Directions: When you describe a graph, name each interval from left to right and say whether the function is increasing, decreasing or constant there, and whether it is linear or nonlinear. Every sketch needs labeled axes.
Part 1: Describe the Graph (Problems 1-3)
A graph of a school bus's distance from school (miles) against time (minutes) goes in a straight line from (0, 0) to (10, 4), stays flat to (12, 4), then goes in a straight line to (20, 9). (a) Describe each interval. (b) Where was the bus moving faster, before or after the stop? Show how you know. (c) What might the flat piece mean?
Two graphs start at (0, 1). Graph P passes through (2, 4), (4, 7) and (6, 10). Graph Q passes through (2, 2), (4, 5) and (6, 10) along a smooth curve. (a) Is each graph increasing or decreasing? (b) Which graph is linear? Explain using the changes in y. (c) On which interval, 0 to 2 or 4 to 6, is Graph Q steeper?
A graph shows the temperature in a town during one spring day: 12 °C at 6 am, 18 °C at 9 am, 22 °C at noon, 24 °C at 3 pm, 20 °C at 6 pm, 16 °C at 9 pm and 13 °C at midnight, joined by a smooth curve. (a) When is the temperature increasing, and when is it decreasing? (b) Is the graph linear? Explain. (c) Between which two readings did the temperature rise fastest?
Part 2: Sketch the Graph (Problems 4-6)
Jada skateboards from home to a friend's house. The first part is uphill, so she goes slowly and steadily. The second part is downhill, and she goes faster, at a steady speed. She stays at her friend's house for 20 minutes, then gets a ride home in a car. Sketch her distance from home against time.
A bathroom sink is filled with water at a steady rate. The tap is turned off and the water sits for a while. Then the plug is pulled, and the water drains quickly at first and more slowly near the end. Sketch the water height against time, and label each piece with its feature words.
A graph is flat at a high value, then decreases in a straight line, then is flat at 0. (a) Write a new story about money that fits this graph (not the lunch account from Guided Practice). (b) Write a second story about something other than money or water. (c) Change your money story so that the decrease slows down near the end. Describe how the sketch changes, using the words linear and nonlinear.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Intervals
Every interval named from left to right
One interval missing or out of order
Intervals not named
Feature Words
Increasing, decreasing, constant, linear and nonlinear used correctly, with reasons
One or two feature words wrong
Feature words missing or mostly wrong
Sketches
Labeled axes; pieces in order with the right shape and steepness
Right order but one shape or steepness wrong
Picture of the place, or pieces missing
Connection to the Situation
Explains what each feature means in the story
Explains some features
No connection to the story
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.
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 does it mean for a function to be increasing on an interval?
Answer: A
Increasing means y goes up as you read the graph from left to right. Choice B describes decreasing. Choice C describes linear, which is a different feature: a straight line can go down. Choice D is about where the graph is, not which way it goes.
Question 2 of 20 · Multiple Choice
A graph goes up in a straight line from (0, 0) to (5, 20), is flat until (8, 20), then goes down in a straight line to (12, 0). On which interval is the function decreasing?
Answer: C
From x = 8 to x = 12 the y-values fall from 20 to 0. Choice A is where the function is increasing. Choice B is where it is constant. Choice D misses the last piece.
Question 3 of 20 · Multiple Choice
A graph is a straight line from (0, 30) down to (6, 0). Which description fits it?
Answer: B
The y-values go from 30 down to 0, so the function is decreasing, and the graph is a straight line, so it is linear: it drops 5 for every 1 in x. Choice A reads the graph from right to left. Choice C calls a straight line nonlinear, but a straight line has a steady rate. Choice D makes both errors.
Question 4 of 20 · Multiple Choice
A smooth graph passes through (0, 2), (1, 3), (2, 5), (3, 9) and (4, 17). Which statement is true?
Answer: D
For each step of 1 in x, y goes up 1, 2, 4 and then 8. The changes are not equal, so the graph curves: it is increasing and nonlinear. Choice A confuses increasing with linear. Choice B looks only at the x-values. Choice C is false: every y-value is larger than the one before.
Question 5 of 20 · Multiple Choice
A graph shows the height of a sunflower over one summer. It rises slowly, then quickly, then levels off. Which statement is true?
Answer: A
Leveling off means the height stops rising, not that it falls, so the height never decreases. The speed of growth changes, so the graph curves and is nonlinear. Choices B and D read "levels off" as "goes down." Choice C ignores the change in speed.
Question 6 of 20 · Multiple Choice
Ana rides her bike away from home along a road. First she pedals up a long hill at a slow, steady speed. Then she coasts down the other side of the hill, going faster and faster. Which graph of her distance from home against time fits?
Answer: B
Ana moves away from home the whole time, so her distance always increases. The slow, steady climb is a straight rise that is not very steep. Coasting faster and faster adds more distance each minute, so the graph curves upward and gets steeper. Choice A draws the shape of the hill instead of the distance. Choice C shows her slowing down. Choice D puts the two parts in the wrong order.
Question 7 of 20 · Multiple Choice
Water is poured at a steady rate into a flask that is wide at the bottom and narrow at the top. Which sketch of water height against time fits?
Answer: D
Near the wide bottom, the water spreads out, so the height rises slowly. Near the narrow top, the same amount of water makes the height rise faster, so the curve gets steeper. Choice B is the graph for a container that is narrow at the bottom. Choice A would fit straight sides only. Choice C shows the water level falling.
Question 8 of 20 · Multiple Choice
A graph of the temperature outside decreases from midnight to 5 am, then increases until noon. Which statement must be true?
Answer: C
The temperature falls until 5 am and rises after, so 5 am is the lowest point between midnight and noon. Choice A may or may not be true; the graph does not say. Choice B cannot be true, since the graph changes direction. Choice D is not given: the lowest point can be any temperature.
Question 9 of 20 · Multiple Choice
A graph of a runner's speed against time is flat from 2 minutes to 10 minutes. What does that mean?
Answer: B
The y-values are speeds, so a flat piece means the speed does not change. Choice A reads the graph as if it showed distance, where flat would mean standing still. Choices C and D would need a piece that goes down or up.
Question 10 of 20 · Multiple Choice
Which set of points lies on a nonlinear graph?
Answer: A
In choice A, y goes up 2, then 4, then 6 for equal steps of 1 in x, so the points cannot lie on one straight line. Choice B goes up 3 each step and choice C goes up 4 for each step of 2, so both are linear. Choice D goes down 3 each step: it is decreasing, but still linear.
Question 11 of 20 · Multiple Choice
Which story matches a graph that increases more and more quickly?
Answer: D
When the number doubles, each hour adds more bacteria than the hour before (for example 100, 200, 400, 800 adds 100, 200, 400), so the graph rises faster and faster: an increasing, nonlinear curve. Choice A grows by the same $5 each week, which is linear. Choice B is constant. Choice C decreases.
Question 12 of 20 · Multiple Choice
A hiker's elevation graph goes in a straight line from (0, 900) to (2, 1500), then in a straight line from (2, 1500) to (5, 1800), with hours and meters. Which statement is true?
Answer: A
The first piece rises 600 m in 2 hours, 300 m per hour. The second rises 300 m in 3 hours, 100 m per hour. So the first piece is steeper. Choice B looks at which piece lasts longer instead of which one rises faster. Choice C ignores the bend at x = 2. Choice D is false: 1,800 m is higher than 1,500 m.
Question 13 of 20 · Multiple Choice
Leo fills a bathtub, turns off the water, and then gets into the tub. Which feature must the graph of the water height show when he gets in?
Answer: C
When Leo gets in, his body pushes the water up, so the height rises quickly even though no water is added. Choice A ignores Leo. Choice B describes draining the tub. Choice D describes a slowing change with no reason in the story.
Question 14 of 20 · Multiple Choice
Two straight lines are drawn on the same axes. Both go up from left to right, and line M is steeper than line N. What does that tell you?
Answer: B
On the same axes, a steeper line has a bigger change in y for each step in x, so line M increases faster. Both lines are straight, so both are linear. Choice A reverses the meaning of steepness. Choice C confuses steep with curved: a steep straight line is still linear. Choice D would need the two lines to be equally steep.
Question 15 of 20 · Short Answer
A graph goes in a straight line from (0, 0) to (3, 6), is flat from (3, 6) to (5, 6), then goes in a straight line down to (9, 0). Describe the graph interval by interval, and say which straight piece is steeper.
0 to 3: increasing and linear (up 2 for each 1 in x). 3 to 5: constant. 5 to 9: decreasing and linear (down 1.5 for each 1 in x). The first piece is steeper, because 2 is more than 1.5.
Question 16 of 20 · Short Answer
A plane waits on the runway, takes off and climbs steadily to its cruising height, flies level for most of the trip, then descends steadily and lands. Sketch its height above the ground against time and describe each piece.
Flat at 0 (waiting), increasing and linear (climbing), constant at the cruising height (the longest piece), decreasing and linear (descending), flat at 0 (landed). Axes: time and height above the ground. A curve at the start or end of the climb is also fine if it is explained.
Question 17 of 20 · Short Answer
A smooth graph passes through (0, 50), (1, 40), (2, 32), (3, 26) and (4, 22). Is it increasing or decreasing? Is it linear or nonlinear? Explain using the changes in y.
Decreasing and nonlinear. For each step of 1 in x, y drops 10, 8, 6 and 4. The drops are not equal, so the graph is a curve, and it falls more slowly as x increases.
Question 18 of 20 · Short Answer
You ride a Ferris wheel for one full turn at a steady speed, starting at the bottom. Sketch your height above the ground against time and describe its features.
The height increases from the lowest point to the highest point halfway through the turn, then decreases back to the lowest point. The graph is nonlinear: it is flattest near the top and bottom and steepest halfway up and halfway down. It looks like one smooth hill.
Question 19 of 20 · Short Answer
Sam sketches his distance from home on a walk to school. His graph has a vertical segment in the middle. Explain why his sketch cannot be right.
A vertical segment gives one time many different distances, so Sam would be in several places at the same moment. That is impossible, and the graph would not be a function (8.F.A.1). A very fast change should be drawn as a steep piece, not a vertical one.
Question 20 of 20 · Short Answer
A graph of the water in a pool goes in a straight line from (0, 0) to (10, 5000), then in a straight line from (10, 5000) to (15, 6000), with hours and gallons. Describe the graph and explain what changed at 10 hours.
Increasing the whole time, made of two linear pieces. The first piece rises 5,000 gallons in 10 hours, 500 gallons per hour. The second rises 1,000 gallons in 5 hours, 200 gallons per hour. At 10 hours the filling slowed down, so the graph becomes less steep.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.F.B.5 mean?
8.F.B.5 means students can read a graph and describe in words how the function behaves, and can draw a graph from a story. They say where the graph goes up, goes down or stays flat, and whether each part is straight (linear) or curved (nonlinear).
Is 8.F.B.5 taught in grade 8 or in Algebra I?
8.F.B.5 is a grade 8 standard in the functions unit of Grade 8 Math. Algebra I builds on it in HSF.IF.B.4, where students also name intercepts (the points where a graph crosses an axis) and the highest and lowest points of a graph. Schools that teach Algebra I in grade 8 often cover both.
What does "qualitative" mean in 8.F.B.5?
It means describing the shape and behavior of a graph in words instead of exact numbers. "The distance goes up quickly, then stays the same" is a qualitative description. Students may use numbers to support it, but the standard is about the features.
How can you tell from a graph whether a function is linear or nonlinear?
Check equal steps in x. If y changes by the same amount for every equal step, the graph is a straight line and the function is linear. If the changes grow or shrink, the graph curves and the function is nonlinear. A graph that only goes up can still be nonlinear.
What does a flat piece on a distance-from-home graph mean?
On a graph of distance from home against time, a flat piece means the distance is not changing, so the person is not moving away or coming closer. What a flat piece means always depends on what the y-axis measures, so students should read the axis label first.
Do sketches for 8.F.B.5 need exact numbers and scales?
No. A sketch must have labeled axes, the pieces in the right order and the right shape and relative steepness. Numbers help when the story gives them, but the standard asks for the qualitative features, not a graph drawn to scale.
What mistakes do students make when describing graphs?
A common one is treating the graph as a picture of the place, for example drawing the shape of a playground slide instead of a graph of the child's height against time. Others are reading right to left, calling every increasing graph linear, and forgetting to label the axes.
How is 8.F.B.5 different from 8.F.B.4?
8.F.B.4 asks for exact linear models: the rate of change, the initial value and an equation. 8.F.B.5 is about the whole shape of a graph, including curved (nonlinear) parts, described in words. Many graphs in 8.F.B.5 have no single equation at all.
How does 8.F.B.5 connect to high school math?
It prepares for HSF.IF.B.4, where students interpret key features of graphs and tables in context, and for HSF.IF.B.6 on average rate of change. Reading a graph as a story is also a big part of science courses, where graphs of motion and temperature are common.
How can parents help with 8.F.B.5 at home?
Sketch graphs of everyday things together: the gas in the car over a week, the brightness outside over a day, or the water in a bottle during a sports practice. Ask your child where the graph goes up, goes down or stays flat, and whether it is straight or curved, and why.
07
Related Standards
6 standards
These standards connect to 8.F.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.A.1Prerequisite
Understand a function as a rule with exactly one output for each input
