Understanding, comparing and constructing functions
What makes a relationship a function, comparing functions across representations, linear vs nonlinear, and building your own
About five lessons of 45 to 60 minutes
A vending machine is a function. A messy junk drawer is not.
Press B4 on a vending machine and you always get the same snack, every single time. That reliability, one input always giving exactly one output, is the entire idea of a function. Compare that to a junk drawer: reach in for 'a pen' and you might pull out three different things. A function is a rule you can trust.
Once you can spot a function, the next skill is reading it in whatever form it shows up: a table, an equation, a graph, or even just a description ('the price starts at $5 and rises 50 cents a day'). Comparing two functions, telling linear from nonlinear, and building your own function from real data are all the same skill applied in different directions.
- A vending machine's button-to-snack rulethe same button always gives the same snack, a real function
- Two phone plans, one shown as an equation, one as a tablecomparing functions across different representations
- A car's distance over time at constant speeda straight-line, linear function
- A balloon's height as it is released and driftsa nonlinear, curving relationship, rising then leveling off
What students will be able to do
Students will determine whether a relationship is a function, compare two functions given in different representations, distinguish linear from nonlinear functions, construct a linear function from a table, graph or verbal description, and describe qualitatively how a function behaves from its graph.
- I can determine whether a table, set of points, or rule represents a function.
- I can compare two functions shown in different forms, such as a table and an equation, by their rate of change and starting value.
- I can tell whether a function is linear (y = mx + b) or nonlinear from a table, equation, or graph.
- I can find the rate of change and starting value from two points or a table, and write the linear function.
- I can describe how a graph behaves, in words: where it increases, decreases, or stays constant.
Standards this unit teaches
- 8.F.A.1Common Core (US)Understand functions
Understand a function as a rule that assigns exactly one output to each input.
- 8.F.A.2Common Core (US)Compare functions
Compare properties of two functions each shown in a different way, such as a table versus a graph.
- 8.F.A.3Common Core (US)Linear versus nonlinear
Recognize that the equation y equals mx plus b defines a straight line and identify functions that are not linear.
- 8.F.B.4Common Core (US)Construct linear functions
Construct a linear function to model a relationship between two quantities and interpret its rate of change.
- 8.F.B.5Common Core (US)Describe functional relationships
Describe qualitatively how a function behaves from its graph, such as where it increases, decreases, or is linear.
Printable anchor chart
Print a wall poster to go with this unit, a code-drawn diagram students can point to during and after the lesson.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 7 function tables & real data graphs worksheetevaluating a rule and reading a rate from a table, the direct foundation for this unit
- Grade 8 linear equations & expressions worksheetsubstituting into a formula, used throughout this unit's worked examples
- Variable in the glossarya function's input and output are both variables, connected by a rule
Words to teach and display
- Function
- a rule that assigns exactly one output to each input
- Input / output
- the value you put into a function's rule, and the single value it produces
- Rate of change
- how much the output changes for each unit increase in the input; the slope of a linear function
- Linear function
- a function of the form y = mx + b, whose graph is a straight line and whose rate of change is constant
- Nonlinear function
- a function whose rate of change is not constant, so its graph is not a straight line
- Initial value
- the output when the input is 0; the y-intercept, b, in y = mx + b
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. What makes a relationship a function
ConcreteA function assigns EXACTLY ONE output to each input. Check a table of pairs by looking only at the input (x) column: if any input value appears twice paired with two DIFFERENT outputs, it is not a function. Outputs are allowed to repeat, only inputs matter.
The table with pairs (1,5), (2,6), (3,7), (3,8) is NOT a function, because the input 3 is paired with two different outputs, 7 and 8. The table (1,5), (2,5), (3,7), (4,9) IS a function, even though the output 5 repeats, because every input still has exactly one output.
Which of these tables represents a function? Table A: (1,5), (2,6), (3,7), (3,8). Table B: (1,5), (2,5), (3,7), (4,9).
- Table A: check each input. The input 3 appears twice, once paired with 7 and once with 8, two different outputs for the same input.
- Table A is NOT a function, since input 3 does not have exactly one output.
- Table B: check each input. Inputs 1, 2, 3, 4 each appear once. The output 5 repeats (for inputs 1 and 2), but that is allowed.
- Table B IS a function, since every input has exactly one output.
Answer: Table A is not a function (input 3 has two outputs). Table B is a function.
- Why is a repeated OUTPUT allowed in a function, but a repeated INPUT with different outputs is not?
- Could a vending machine that sometimes gives a different snack for the same button still be called a function?
2. Comparing functions across representations
PictorialTwo functions can be compared even when they are shown completely differently, one as an equation, one as a table, one as a graph, as long as you can find each one's rate of change and starting value (initial value).
Function A is given by the equation y = 3x + 1: its rate of change is 3 (the coefficient of x) and its initial value is 1 (when x = 0). Function B is given by the table (0,5), (1,7), (2,9): its rate of change is (7-5)/(1-0) = 2, confirmed by (9-7)/(2-1) = 2, and its initial value is 5 (the output when x = 0). Function A grows faster (rate 3 vs 2) but starts lower (1 vs 5).
Function A: y = 3x + 1. Function B, given by the table (0,5), (1,7), (2,9). Compare their rates of change and initial values.
- Function A's rate of change is the coefficient of x: 3. Its initial value (at x = 0) is 1.
- Function B's rate of change from the table: (7-5)/(1-0) = 2/1 = 2, and (9-7)/(2-1) = 2/1 = 2, confirming a constant rate of 2.
- Function B's initial value is its output at x = 0, which is 5 directly from the table.
- Compare: A has the larger rate of change (3 > 2), so A grows faster, but B has the larger initial value (5 > 1), so B starts higher.
Answer: Function A's rate of change is 3, initial value 1. Function B's rate of change is 2, initial value 5. A grows faster; B starts higher.
- Why do you need at least two points from a table to find its rate of change?
- If two functions have the same rate of change but different initial values, will their graphs ever cross?
3. Linear versus nonlinear functions
PictorialA function is linear exactly when it matches the form y = mx + b, and its graph is always a straight line. The fastest way to check a table is to look at the DIFFERENCES between consecutive outputs: if they are constant, the function is linear; if they change, it is nonlinear.
The table (0,2), (1,5), (2,8), (3,11) has output differences 3, 3, 3, always the same, so it is linear with rate of change 3. The table (0,1), (1,2), (2,4), (3,8) has output differences 1, 2, 4, which change every time, so it is nonlinear (this one doubles each step, an exponential relationship, not a straight line).
- What single check on a table's outputs tells you whether a function is linear?
- Could two different nonlinear functions have the exact same first two output differences? What would you check next?
4. Constructing a linear function
AbstractTo build a linear function from two known points (or a rate and a starting value described in words), find the rate of change m first, then use one point to solve for the initial value b in y = mx + b.
A plant is 4 cm tall at week 1 and 10 cm tall at week 4. The rate of change is (10-4)/(4-1) = 6/3 = 2 cm per week. Using the point (1,4): 4 = 2(1) + b, so b = 2. The function is height = 2w + 2, meaning the plant started at 2 cm (week 0) and grows 2 cm every week.
A candle is 20 cm tall when lit and burns down at a constant rate of 2 cm per hour. Write the linear function for its height, and find its height after 5 hours.
- The rate of change is -2 cm per hour (the height DEcreases, so the rate is negative): m = -2.
- The initial value (at t = 0 hours) is the starting height, 20 cm: b = 20.
- Write the function: height = -2t + 20.
- Substitute t = 5: height = -2(5) + 20 = -10 + 20 = 10.
Answer: height = -2t + 20. After 5 hours, the candle is 10 cm tall.
- Why is the rate of change negative in the candle problem?
- If you only knew two points on a line, not a verbal description, how would you find b?
5. Describing a function's behaviour from its graph
AbstractNot every function needs numbers to describe it. Looking at a graph's shape, you can describe where it increases, decreases, stays flat, or curves, in plain words, exactly the skill needed to sketch a graph from a story or read the story back out of a graph.
A hiker's elevation over a 3-hour hike: climbing steadily for the first hour, walking along a flat ridge for the second hour, then descending quickly in the third hour. Each of those three behaviours, increasing, constant, then decreasing, matches a distinct piece of the graph.
- Which part of the elevation graph shows the hiker was not gaining or losing height?
- How would the graph look different if the hiker had descended slowly instead of quickly in hour 3?
Common misconceptions and how to address them
MisconceptionA table is not a function if any output value repeats.
Why it happens: Students confuse the rule (each INPUT needs one output) with a stricter, incorrect rule about outputs never repeating.
How to address it: Only check the INPUT column. Outputs are allowed to repeat freely; a function only breaks when one input is paired with two different outputs. Revisit the vending-machine hook: many buttons could give the same snack, and it would still be a function.
MisconceptionEvery function is a straight line.
Why it happens: Linear functions are usually the first kind students meet in depth, so 'function' and 'straight line' get bundled together in memory.
How to address it: Show a clearly nonlinear function, such as y = x^2 or the doubling table in section 3, and confirm it still fits the definition (one output per input) while curving instead of forming a line. 'Function' is the broad category; 'linear' is one type within it.
MisconceptionYou can only find a function's rate of change from its graph.
Why it happens: Students overly associate slope with a picture, and struggle to compute it from a table or an equation.
How to address it: The rate of change can be found from ANY representation: read it directly as the coefficient of x in an equation, or compute (change in y)/(change in x) from any two rows of a table. Practise finding it from all three forms side by side, as in section 2.
MisconceptionIn a word problem, the FIRST number mentioned is always the rate of change.
Why it happens: Students latch onto the order words appear in, rather than checking whether each number describes a per-unit change or a fixed starting amount.
How to address it: Identify which quantity changes with the input (the rate) and which is fixed at the start (the initial value) by asking 'does this number change as time/quantity increases?' In the candle problem, 20 cm is fixed at the start, while 2 cm per hour is the rate that keeps applying.
MisconceptionOn a graph where the x-axis is time, 'increasing' means time is moving forward.
Why it happens: Since time always moves forward left to right, students confuse the x-axis's steady increase with the function itself increasing.
How to address it: 'Increasing' and 'decreasing' always describe the OUTPUT (the y-value), not the input. Point to a specific stretch of the hiker graph and ask 'is the elevation going up or down here', not 'is time going up here' (time is always going up, left to right, on every such graph).
Guided practice (with answers)
1. Does the table (1,3), (2,3), (3,3), (4,3) represent a function?
Answer: Yes, because every input (1, 2, 3, 4) has exactly one output, even though the output is always 3.
2. Function A: y = 5x - 2. Function B is given by the table (0,1), (1,4), (2,7). Which has the greater rate of change?
Answer: Function A, because its rate of change is 5, while Function B's rate of change is (4-1)/(1-0) = 3.
3. Is the table (0,0), (1,1), (2,4), (3,9) linear or nonlinear?
Answer: Nonlinear, because the output differences are 1, 3, 5 (not constant); this table is actually y = x^2.
4. A savings account starts with $50 and grows by $10 a week. Write the linear function, using w for weeks.
Answer: savings = 10w + 50, because the rate of change is 10 (per week) and the initial value is 50.
5. A graph shows a function rising steeply from x=0 to x=2, then staying perfectly flat from x=2 to x=4. Describe what happens between x=2 and x=4.
Answer: The function is constant (neither increasing nor decreasing) between x=2 and x=4; the output stays the same value across that whole interval.
Independent practice worksheets
Practise identifying, comparing, and constructing functions with computed, never-wrong answer keys.
Differentiation
- Start the function-or-not check with tables where the repeated input is easy to spot (listed right next to each other) before scattering the rows out of order.
- Provide a simple two-column 'rate of change / initial value' template for comparing two functions, so the comparison stays organised.
- Colour-code the output differences directly on a table (green if constant, red if not) to make the linear/nonlinear check visual.
- For constructing functions, always write 'rate = ___, start = ___' as a scaffold before writing the equation itself.
- Compare three functions at once, given in three different representations (equation, table, graph description), ranking them by rate of change and by initial value separately.
- Investigate a function whose rate of change looks constant for the first three rows of a table but then changes, to reinforce checking every pair, not just the first.
- Construct a linear function from a real data set with imperfect (rounded) numbers, discussing why the 'best' rate might not come from just the first two points.
- Sketch a graph from a more complex verbal description with four or five distinct behaviours (increasing fast, increasing slow, flat, decreasing, flat again).
Assessment: exit ticket
A three-question exit ticket sampling function identification, comparison, and construction.
1. Does the table (2,7), (3,9), (2,11), (4,13) represent a function? Why or why not?
Answer: No, because the input 2 appears twice with two different outputs, 7 and 11.
2. Function A has rate of change 4. Function B is given by the table (0,2), (1,5), (2,8). Which grows faster?
Answer: Function A, because its rate of change is 4, while Function B's rate of change is (5-2)/(1-0) = 3 (confirmed by (8-5)/(2-1) = 3), and 4 is greater than 3.
3. A gym charges a $30 sign-up fee plus $15 per month. Write the linear function using m for months, and find the cost after 6 months.
Answer: cost = 15m + 30. After 6 months: 15(6) + 30 = 90 + 30 = 120, so $120.
Teacher notes and timings
- Rough timing across five lessons: Lesson 1 what is a function (section 1), Lesson 2 comparing functions (section 2), Lesson 3 linear vs nonlinear (section 3), Lesson 4 constructing linear functions (section 4), Lesson 5 qualitative description plus the exit ticket (section 5 and assessment).
- This is a five-standard unit (8.F.A.1 through 8.F.B.5), the full Grade 8 Functions domain, taught as one deliberate progression: define a function, compare functions, classify linear vs nonlinear, build one, then read one qualitatively. Each section leans on the one before it.
- This unit assumes comfort with substituting into a linear rule (Grade 8 linear equations unit) and reading a rate from a table (Grade 7 function tables unit). Revisit either first if constructing functions in section 4 feels shaky.
- Language to repeat: a function needs exactly one output per input (never the other way around); 'linear' means constant rate of change; 'increasing/decreasing' always describes the output, never the input.
- The exit ticket's second question deliberately has a nearby wrong-seeming rate (4 vs 3) to check students compute Function B's rate of change from the table rather than guessing from the numbers' appearance.
- Print and present both work: use the Print button for a clean handout, or work the hiker elevation story live on the board, asking students to sketch it themselves before revealing the actual graph.