8.F.A.3: Linear Functions and Functions That Are Not Linear
In plain English: 8.F.A.3 is the Common Core grade 8 math standard that asks students to read the equation y = mx + b as a linear function whose graph is a straight line, and to give examples of functions that are not linear, such as the area of a square, A = s². Students test tables, graphs and equations for equal steps. It is part of the functions unit of Grade 8 Math.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Common Core State Standards for Mathematics · Domain: Functions (F) · Cluster: Define, evaluate, and compare functions. Also written as 8.F.3 · Official standard
Students already know that a function gives each input exactly one output (8.F.A.1), and they have seen the slope m and the equation y = mx + b of a line (8.EE.B.6). This lesson puts the two ideas together. Every equation of the form y = mx + b defines a linear function: a function whose graph is a straight line. Its outputs change by the same amount, m, each time the input goes up by 1, and b is the output when the input is 0. A nonlinear function is any function whose graph is not a straight line.
The official example is the area of a square, A = s² (s squared, meaning s times s). Its graph contains (1, 1), (2, 4) and (3, 9), and these three points are not on one straight line; Example 2 and Diagram 1 work through it. Students then build their own examples of functions that are not linear from areas, repeated doubling and sharing a fixed cost. The grade 8 functions standards carry the note "Function notation is not required in Grade 8," so every function on this page is written in words or as an equation such as y = 3x + 2.
Learning Objectives
By the end of this lesson, students will be able to:
Explain why an equation y = mx + b defines a linear function whose graph is a straight line
Name the slope m and the value b of a linear equation and describe what each tells about its graph
Decide whether a table, a graph, an equation or a situation shows a linear function, and justify the decision
Give examples of functions that are not linear and show with three points that their graphs are not straight lines
Prior Knowledge Required
Students should already be comfortable with:
Knowing that a function gives each input exactly one output 8.F.A.1
Using the slope m and deriving the equation y = mx + b of a line 8.EE.B.6
Evaluating expressions with whole-number exponents, such as 3² 6.EE.A.1
Plotting ordered pairs, including negative ones 6.NS.C.8
Give each pair a sheet of grid paper. Read the prompt aloud.
Warm-Up Prompt
"Draw squares with sides of 1, 2 and 3 grid units. For each square, write its perimeter and its area. If you add 1 to the side, how much does the perimeter grow each time? How much does the area grow each time?"
Collect answers in a table on the board. The perimeters are 4, 8 and 12, so the perimeter grows by 4 every time. The areas are 1, 4 and 9, so the area grows by 3 and then by 5. Ask: which quantity do you think will graph as a straight line? Keep the table on the board; the class returns to it in Example 2.
Direct Instruction20 minutes
Define each term and write it on an anchor chart:
Linear function: a function whose graph is a straight line. Every equation y = mx + b defines one.
m, the slope: the change in y for each increase of 1 in x. It is the same between any two points, which is why the graph is straight.
b: the value of y when x = 0. The line crosses the y-axis at (0, b).
Nonlinear function: a function whose graph is not a straight line. Its outputs do not change by the same amount for equal steps in the input.
Three-point test: three points lie on one straight line only if the slope from the first to the second equals the slope from the second to the third. Work out the slope as (change in y) ÷ (change in x).
Work through the examples. Ask students to predict "linear or not" before each one is worked out.
Reading y = mx + b in a context
A 20-liter jug holds 2 liters of water. A tap adds 3 liters each minute, so after x minutes the jug holds y = 3x + 2 liters. Make a table for x = 0 to 4 and describe the graph.
Equation: Pairs (0, 2), (1, 5), (2, 8), (3, 11), (4, 14). Each minute adds 3 liters, so the points lie on a straight line. m = 3 is the slope (liters per minute), and b = 2 is where the line crosses the y-axis (the starting amount). The jug is full at x = 6, when y = 20.
The official example: area of a square
The area of a square with side length s is A = s². Is this function linear?
Equation: The graph contains (1, 1), (2, 4) and (3, 9). The slope from (1, 1) to (2, 4) is 3, and from (2, 4) to (3, 9) it is 5. The slopes differ, so the points are not on one straight line and A = s² is not linear.
Which equations are linear?
Decide which of these define linear functions: y = 4 - x, y = x ÷ 2, y = 7, y = x² + 1, y = 12 ÷ x.
Equation: Linear: y = 4 - x (m = -1, b = 4), y = x ÷ 2 (m = 1/2, b = 0) and y = 7 (m = 0, b = 7, a horizontal line). Not linear: y = x² + 1 gives 1, 2, 5 for x = 0, 1, 2 (steps 1 and 3), and y = 12 ÷ x gives 12, 6, 4 for x = 1, 2, 3 (steps -6 and -2).
A table with uneven input steps
Is this table linear? x = 0, 2, 5, 6 gives y = 11, 7, 1, -1.
Equation: Slopes: (7 - 11) ÷ 2 = -2, (1 - 7) ÷ 3 = -2, (-1 - 1) ÷ 1 = -2. Every slope is -2, so the table is linear: y = -2x + 11. Compare slopes, not raw output steps, when the inputs skip.
A nonlinear example from real life
A rectangle has an area of 12 square centimeters. Its length is a function of its width: length = 12 ÷ width. Use the widths 1, 2, 3, 4 and 6 cm.
Equation: The lengths are 12, 6, 4, 3 and 2 cm. From width 1 to 2 the length drops by 6, from 2 to 3 it drops by 2, and from 3 to 4 by 1. The changes are not equal, so this function is not linear.
Use Diagram 2 with Example 1: every step of 1 to the right goes up 3, so the steps line up in a straight line. Use Diagram 1 with Example 2: the dashed line through (1, 1) and (2, 4) would reach (3, 7), but the area of the 3-by-3 square is 9. Stress the logic of the standard in both directions: y = mx + b always gives a straight line, and a function whose graph bends cannot be written as y = mx + b.
Guided Practice15 minutes
Pairs work on four items, one at a time. After each one, a pair explains its answer at the board.
Make a table for y = -2x + 6 with x = 0, 1, 2, 3. Is it linear? Name m and b. (Outputs 6, 4, 2, 0. Linear, since each step goes down 2. m = -2 and b = 6, so the line crosses the y-axis at (0, 6).)
x = 1, 2, 3, 4 gives y = 5, 8, 13, 20. Is the function linear? (No. The steps are 3, 5 and 7.)
The perimeter of a square is P = 4s. Is it linear? (Yes: m = 4 and b = 0. Compare it with the area in Example 2.)
Do (0, 2), (2, 5) and (4, 8) lie on one line? What about (0, 2), (2, 5) and (4, 9)? (The first set does: both slopes are 1.5. The second does not: the slopes are 1.5 and 2.)
Listen for students who call every equation with an x linear, and for students who test only two points. Any two points lie on some line; it takes a third point to show a function is not linear.
Independent Practice15 minutes
Students work alone on four items, then compare with a partner:
Linear or not? For each linear one, give m and b: y = 5x - 2, y = x² - 3, y = -x, y = 10. (Linear: y = 5x - 2 with m = 5, b = -2; y = -x with m = -1, b = 0; y = 10 with m = 0, b = 10. Not linear: y = x² - 3.)
x = 0, 1, 2, 3 gives y = 40, 20, 10, 5. Is it linear? (No. The steps are -20, -10 and -5.)
Which of these points are on the line y = 0.5x + 3: (4, 5), (6, 7), (-2, 2)? ((4, 5) and (-2, 2) are on it. (6, 7) is not, since 0.5(6) + 3 = 6.)
Invent a nonlinear function from your own life. Write a table with three points and show that the slopes differ.
Closure5-10 minutes
Exit ticket: (1) Is y = 3 - 4x linear? If so, name m and b. (Yes: m = -4 and b = 3.) (2) Write one function that is not linear and give three of its points that are not on a line. (3) Finish the sentence: "The graph of y = mx + b is a straight line because ..."
Differentiation Strategies
For Struggling Students
Give a table template with a third column labeled "change in y" so students write each step before they decide
Start with tables whose inputs go up by 1, and add tables with skipped inputs only after students can compute a slope
Let students test straightness with a ruler on their plotted points before they compute slopes
For Advanced Students
Rewrite 2x + y = 9 in the form y = mx + b and explain why it defines a linear function
Explain why the doubling pattern 1, 2, 4, 8 can never be written as y = mx + b, however m and b are chosen
Preview (beyond this standard): in high school, functions such as y = x² and y = 2 to the power x get their own names, quadratic and exponential
Assessment Guidance
What to Look For
A strong answer uses equal changes: "each step of 1 in x adds 3 to y, so the graph is a straight line." For nonlinear functions, look for three points and two different slopes, not two points. Watch for four errors: calling any equation with x linear, thinking y = mx + b must have a nonzero b, comparing output steps when the inputs skip, and checking only two points. Ask students what m and b mean in context, not only their values.
02
Classroom Activities
3 Activities
1
Linear or Not? Card Sort
15 minPairs
Pairs sort 10 cards into two piles, "Linear" and "Not linear." The cards use equations, tables, points on a graph and situations. For every linear card, pairs also write m and b.
The 10 Cards (with answers)
Card 1, equation: y = 6x - 4 (linear: m = 6, b = -4)
Card 2, equation: y = x² (not linear)
Card 3, equation: y = -0.5x + 9 (linear: m = -0.5, b = 9)
Card 5, table: x = 0, 2, 4, 6 gives y = 10, 7, 4, 1 (linear: slope -1.5, b = 10)
Card 6, situation: the cost of apples at $2 per pound (linear: m = 2, b = 0)
Card 7, situation: the number of handshakes when everyone in a group shakes hands once with everyone else, as a function of the group size (not linear)
Card 8, situation: a bike share costs $3 to unlock plus $0.25 per minute (linear: m = 0.25, b = 3)
Card 9, graph: the points (0, 1), (1, 3), (2, 9) (not linear: slopes 2 and 6)
Card 10, equation: y = 20 ÷ x (not linear)
Procedure
For each card, write the reason on the back: "equal steps of ..." or "slopes ... and ... differ"
For each equation card, make a quick table with three inputs before deciding
When both piles are done, check with another pair and settle disagreements with a three-point test
Discussion Questions
Five cards are linear and five are not. Which card was hardest to decide, and why?
Card 6 and Card 8 are both linear. Which one has b = 0, and what does b mean for the bike share?
Card 7 has no numbers. How can you test it? (Try groups of 2, 3 and 4 people: 1, 3 and 6 handshakes, so the steps are 2 and 3.)
Modification for Distance Learning
Put the cards on a shared slide with two labeled boxes. Pairs drag each card into a box and type their three-point test beside it.
2
Squares on Grid Paper
20 minPairs
Pairs extend the warm-up. They draw squares with sides of 1 to 5 cm on centimeter grid paper, record the perimeter and the area of each, and graph both functions on the same axes.
Procedure
Draw squares with sides 1, 2, 3, 4 and 5 cm. Count the unit squares inside each one to find its area
Record a table: side s, perimeter P and area A
Plot the points (s, P) in one color and the points (s, A) in another. Lay a ruler along each set of points
Write the change in P and the change in A for each step of 1 cm
Expected Results
Perimeters: 4, 8, 12, 16, 20 cm, which grow by 4 each step. Areas: 1, 4, 9, 16, 25 square cm, which grow by 3, 5, 7 and 9. The perimeter points lie along the ruler, P = 4s. The area points curve upward, so A = s² is not linear.
Discussion Questions
The area changes grow by 2 every step. What would the next change be for a 6 cm square?
Which equation fits the perimeter points? Why can no equation y = mx + b fit the area points?
Where do the two graphs cross, and what is special about that square?
Challenge Variation
Pairs draw squares with sides of 2, 3, 4 and 5 cm and count the border squares: the unit squares that touch an edge of the big square. Is the number of border squares a linear function of the side length? (Yes: 4, 8, 12 and 16, which grow by 4 each step, so the rule is 4s - 4.)
3
Tear or Fold
20 minPairs
Two actions with a sheet of scrap paper give two functions. Tearing a piece off a strip adds one piece each time. Folding a sheet in half doubles the number of layers each time. Pairs record both, graph both and decide which one is linear.
Procedure
Partner 1 tears a strip of paper into pieces, one tear at a time, and records (number of tears, number of pieces) for 0 to 5 tears
Partner 2 folds a second sheet in half again and again and records (number of folds, number of layers) for 0 to 5 folds, counting the layers at the edge
Both partners plot their points on one grid and compute the change in the output for each step
Write an equation for the tearing function
Expected Results
Tearing: 0 to 5 tears give 1 to 6 pieces, one more each time, so pieces = tears + 1, a linear function with m = 1 and b = 1. Folding: 0 to 5 folds give 1, 2, 4, 8, 16 and 32 layers. The changes are 1, 2, 4, 8 and 16, so the folding function is not linear.
Discussion Questions
After how many folds does the folding function first have more layers than the tearing function has pieces?
Why does the folding graph get steeper while the tearing graph keeps the same steepness?
Most people cannot fold a sheet more than 6 or 7 times. Why does the number of layers explain this?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Official Example, A = s² Is Not Linear
The graph of A = s² for sides from 0 to 4, drawn to scale. The dashed line through (1, 1) and (2, 4) has slope 3 and would pass through (3, 7), but the area of a square with side 3 is 9. The slope from (2, 4) to (3, 9) is 5, so the three points are not on a straight line.
Diagram 2: Why y = mx + b Graphs as a Straight Line
The jug from Example 1, y = 3x + 2, drawn to scale. Each step of 1 minute to the right goes up the same 3 liters (m = 3), so the steps line up along one straight line. The line crosses the y-axis at (0, 2), since b = 2, and reaches 20 liters, a full jug, at 6 minutes.
04
Homework Assignment
~30 min
8.F.A.3 Homework: Linear and Nonlinear Functions
Directions: Show your tables and your slope calculations. To show that a function is not linear, give three points and two different slopes.
Part 1: Reading y = mx + b (Problems 1-3)
For y = -4x + 10: (a) Make a table for x = 0, 1, 2 and 3. (b) How much does y change for each step of 1 in x? (c) Name m and b, and give the point where the graph crosses the y-axis.
Decide whether each equation defines a linear function. For each linear one, give m and b. (a) y = 8 - 3x (b) y = 2x² - 1 (c) y = x ÷ 4 + 1 (d) y = 9 ÷ x (e) y = -6
A plumber charges C = 75h + 50 dollars for a visit that lasts h hours. (a) Is C a linear function of h? Explain. (b) What do 75 and 50 mean in this situation? (c) What does a 3-hour visit cost?
Part 2: Linear or Not? (Problems 4-6)
Decide whether each table shows a linear function and explain. (a) x = 0, 1, 2, 3, 4 gives y = 5, 9, 13, 17, 21. (b) x = 1, 2, 3, 4, 6 gives y = 24, 12, 8, 6, 4. (c) x = 0, 3, 6, 9 gives y = 2, 4, 6, 8.
The area of a circle with radius r is A = πr². Use 3.14 for π. (a) Find the area for r = 1, 2 and 3 cm. (b) Use the slopes between these points to show that the graph is not a straight line.
(a) Ben says y = x² - x is linear, because (0, 0) and (1, 0) are both on its graph and both lie on the line y = 0. Use a third point to show that Ben is wrong. (b) Describe one function from real life that is not linear, and give three of its points that show it.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Tables and Values
All tables and outputs are correct
One or two arithmetic errors
Many errors or missing tables
Linear Decisions
Every linear or nonlinear decision is correct
One or two decisions are wrong
Most decisions are wrong
Reasons
Uses equal steps or three points with two slopes
Reason given but incomplete, for example only two points
No reasons
Meaning of m and b
Names m and b and explains them in context
Names m and b without meaning
m and b missing or wrong
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
Which equation defines a linear function?
Answer: B
y = 5 - 0.5x is the same as y = -0.5x + 5, so m = -0.5 and b = 5, and its graph is a straight line. Choice A squares x: x = 0, 1, 2 give 0, 3, 12, steps of 3 and 9. Choice C gives 10, 5 and 10/3 for x = 1, 2, 3, which are unequal steps. Choice D gives 1, 2, 9 for x = 0, 1, 2.
Question 2 of 20 · Multiple Choice
What does the equation y = -2x + 7 tell you about its graph?
Answer: D
Here m = -2, so y goes down 2 for each step of 1 to the right, and b = 7, so the line crosses the y-axis at (0, 7). Choice A ignores the minus sign. Choice C swaps m and b, and choice B swaps them but keeps the minus sign on the slope.
Question 3 of 20 · Multiple Choice
Each table uses the inputs x = 0, 1, 2, 3. Which list of outputs shows a linear function?
Answer: B
Choice B goes up by 3 every step, so it is linear. Choice A triples (steps 2, 6, 18). Choices C and D change by 1, 2, 3 and by -1, -2, -3: the outputs always go up or always go down, but not by equal amounts.
Question 4 of 20 · Multiple Choice
The volume of a cube with edge length s is V = s³. Its graph contains (1, 1), (2, 8) and (3, 27). What are the slopes from (1, 1) to (2, 8) and from (2, 8) to (3, 27), and what do they show?
Answer: A
(8 - 1) ÷ (2 - 1) = 7 and (27 - 8) ÷ (3 - 2) = 19. The slopes differ, so the points are not on one line. Choice B divides 8 by 2 - 1 without subtracting the first output. Choice D uses the outputs 8 and 27 as if they were slopes. Choice C repeats the first slope without computing the second.
Question 5 of 20 · Multiple Choice
Which point is on the graph of y = 4x - 3?
Answer: C
For x = 4, y = 4(4) - 3 = 13. Choice A swaps the input and the output: x = 13 gives 49, not 4. Choice B drops the minus sign of b: x = 0 gives -3, not 3. Choice D subtracts in the wrong order, 3 - 4 = -1: x = 1 gives 4(1) - 3 = 1.
Question 6 of 20 · Multiple Choice
Which situation is NOT described by a linear function?
Answer: D
A thrown ball rises, slows down, stops and falls back, so its height goes up and then down. For example, thrown up at 15 meters per second, and taking gravity as about 10 meters per second each second (g ≈ 10 m/s²), its height above the hand is about 0, 10, 10 and 0 meters after 0, 1, 2 and 3 seconds: steps of 10, 0 and -10, not equal. Choices A, B and C each add the same amount per unit: y = 12x, y = 2x + 5 and y = 1.4x.
Question 7 of 20 · Multiple Choice
The points (0, 4), (2, 10) and (5, k) lie on one straight line. What is k?
Answer: B
The slope is (10 - 4) ÷ 2 = 3, so the line is y = 3x + 4 and k = 3(5) + 4 = 19. Choice A adds only two steps of 3 to 10. Choice D adds one step of 3 to 10. Choice C multiplies 3 by 5 and adds 10 instead of 4.
Question 8 of 20 · Multiple Choice
A line has slope 0 and crosses the y-axis at (0, 6). Which equation defines this linear function?
Answer: A
With m = 0 and b = 6, y = 0x + 6, which is y = 6: a horizontal line. Choice B is a vertical line, which is not a function. Choice C has slope 6 and passes through (0, 0). Choice D has slope 1.
Question 9 of 20 · Multiple Choice
Write y = (x + 8) ÷ 2 in the form y = mx + b. Is it linear?
Answer: C
Divide both terms by 2: y = x ÷ 2 + 8 ÷ 2 = 0.5x + 4. So m = 0.5 and b = 4. Choice A divides only x by 2. Choices B and D judge the look of the equation instead of its form: it can be written as y = mx + b, so it is linear.
Question 10 of 20 · Multiple Choice
A table shows x = 0, 1, 2, 3, 4 and y = 2, 5, 8, 12, 15. Which statement is true?
Answer: A
The steps are 3, 3, 4 and 3. One unequal step is enough to break the straight line. Choice B checks only the first steps. Choice C confuses "always increasing" with "increasing by the same amount."
Question 11 of 20 · Multiple Choice
A family drives 120 miles. The time in hours is t = 120 ÷ s, where s is the average speed in miles per hour. Is t a linear function of s?
Answer: B
Speeds of 20, 30 and 40 mph give 6, 4 and 3 hours. Equal steps of 10 mph cut the time by 2 hours, then by 1 hour, so the graph is not a straight line. Choice A is true but does not prove linearity: many decreasing functions are curved. Choice C looks at the look of the equation, and choice D is not a reason at all.
Question 12 of 20 · Multiple Choice
Which set of points could be on the graph of a linear function?
Answer: C
In choice C every step of 2 in x lowers y by 1, so every slope is -0.5. Choice A doubles, with slopes 1, 2 and 4. Choice B has slopes -3, -1 and -1/3. Choice D has slopes 1, 2 and 1: it looks almost straight, but one step breaks the pattern.
Question 13 of 20 · Multiple Choice
In y = 2x + b, the value of b changes from 2 to 5 and nothing else changes. What happens to the graph?
Answer: D
Every output grows by 3, since 2x + 5 is 3 more than 2x + 2, so each point moves up 3 and the slope stays 2. Choice A confuses b with the slope. Choice C forgets that y = mx + b is always a straight line. Choice B gets the direction wrong: moving up 3 is the same as moving left 1.5 here, not right.
Question 14 of 20 · Multiple Choice
Jada says that y = 2x is not a linear function because it has no b. Which statement is correct?
Answer: A
y = 2x is y = 2x + 0, so m = 2 and b = 0, and its line passes through the origin, (0, 0). Choice C reads the slope 2 as b. Choices B and D treat b = 0 as "no b", but 0 is an allowed value.
Question 15 of 20 · Short Answer
Make a table for y = 0.5x - 1 with x = -2, 0, 2 and 4. Is the function linear? Give its slope and the point where its graph crosses the y-axis.
Outputs: 0.5(-2) - 1 = -2, then -1, 0 and 1. Linear: each step of 2 in x adds 1 to y, so the slope is 0.5, and the graph crosses the y-axis at (0, -1).
Question 16 of 20 · Short Answer
Use the points for x = 0, 1 and 2 to show that y = x² + 2 is not a linear function.
The points are (0, 2), (1, 3) and (2, 6). The slopes are (3 - 2) ÷ 1 = 1 and (6 - 3) ÷ 1 = 3. The slopes differ, so the points are not on one line and the function is not linear.
Question 17 of 20 · Short Answer
A linear function's graph crosses the y-axis at (0, -3) and goes up 5 for each 1 unit to the right. Write its equation. Is the point (2, 7) on its graph?
y = 5x - 3. For x = 2, y = 5(2) - 3 = 7, so yes, (2, 7) is on the graph.
Question 18 of 20 · Short Answer
A sunflower is 6 cm tall and grows 1.5 cm each day. Write an equation for its height h after d days. Is h a linear function of d? What do the two numbers in your equation mean?
h = 1.5d + 6. It is linear, of the form y = mx + b. The slope 1.5 is the growth in centimeters per day, and 6 is the height in centimeters on day 0.
Question 19 of 20 · Short Answer
Give an example of a function from real life that is not linear. Write a rule, three input-output pairs and the two slopes that show it is not linear.
Answers vary. Sample: a $60 pizza order shared equally: cost per person = 60 ÷ p. For p = 1, 2, 3 people the cost is $60, $30 and $20. The slopes are -30 and -10, which differ, so the function is not linear.
Question 20 of 20 · Short Answer
Do the points (1, 2), (3, 8) and (6, 17) lie on one straight line? Explain, and write the equation of the line if they do.
Yes. The slopes are (8 - 2) ÷ (3 - 1) = 3 and (17 - 8) ÷ (6 - 3) = 3, which are equal. Going back 1 unit from (1, 2) gives b = 2 - 3 = -1, so the line is y = 3x - 1.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.F.A.3 mean?
8.F.A.3 means students understand that every equation y = mx + b defines a linear function, one whose graph is a straight line. They also give examples of functions that are not linear, such as the area of a square, and show that their graphs are not straight lines.
Is 8.F.A.3 taught in grade 8 or in Algebra I?
8.F.A.3 is a grade 8 standard in the functions unit, usually taught right after students learn the slope and y = mx + b in 8.EE.B.6. Schools that teach Algebra I in grade 8 cover it early in that course, and high school builds on it in HSF.LE.A.1.
How can you tell if a function is linear from a table?
Check that equal steps in x give equal steps in y. If the inputs skip, compare slopes instead: divide each change in y by its change in x. For example, x = 0, 3, 4 with y = 1, 7, 9 has slopes 6 ÷ 3 = 2 and 2 ÷ 1 = 2, so it is linear.
Why is A = s² not a linear function?
Because its graph bends. The official example uses the points (1, 1), (2, 4) and (3, 9): the slope is 3 between the first two and 5 between the last two. Points on one straight line always have equal slopes, so these points are not on a line.
Is a horizontal line a linear function?
Yes. A horizontal line has slope 0, so its equation is y = 0x + b, usually written y = b. Every input has the same output. A vertical line, such as x = 4, is not a function at all, because one input has many outputs.
What are good examples of nonlinear functions for grade 8?
Areas of squares and circles as functions of their sides or radii, a quantity that doubles each step (folding paper, a savings account that grows by the same percent), and sharing a fixed cost among more people. Each one has unequal output steps for equal input steps.
Do students need f(x) notation for 8.F.A.3?
No. The grade 8 functions standards carry the footnote "Function notation is not required in Grade 8." Students write equations such as y = 3x + 2 or A = s². Function notation comes in high school, in HSF.IF.A.2.
What mistakes do students make with linear and nonlinear functions?
Many students think any equation with an x is linear, or that y = mx + b must have a b that is not 0. Others check only two points, which always lie on some line, or compare output steps in a table whose inputs skip. A three-point slope test fixes all of these.
How does 8.F.A.3 connect to high school math?
In Algebra I, HSF.LE.A.1 asks students to tell linear functions (equal differences) from exponential ones (equal factors), and HSF.IF.C.7 asks them to graph linear, quadratic and other functions. The grade 8 idea of equal steps is the base for both.
How can parents help with 8.F.A.3 at home?
Look for patterns together. A phone bill with a fixed fee plus a price per gigabyte is linear; the area of a square rug as its side grows is not. Ask your child to make a small table for each and to explain which one would graph as a straight line.
07
Related Standards
6 standards
These standards connect to 8.F.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.EE.B.6Prerequisite
Use similar triangles to explain slope and derive y = mx and y = mx + b
