One steady change, four ways to show it

Explain it like I am five

Imagine you watch a train from a station. When you start your clock, the train is already 250 meters away. Each second, it moves another 83 meters away. The seconds you enter are the input. The distance you get back is the output. A function is a rule that gives one output for each allowed input. This function has a constant rate of change: equal amounts of time add equal amounts of distance. You can tell that same story with words, a formula, a table, or a graph. These are four views of one relationship, like four descriptions of the same trip.

The input runs horizontally and the output runs vertically.

Reminder

Function notation. f(2) means output at input 2: for f(x) = 2x + 1, f(2) = 5.
Multiply before adding. 83 × 2 + 250 = 166 + 250 = 416.
Coordinate order. (2, 416) places input seconds first and output meters second.
Signed multiplication. 2(−1) = −2 because multiplying a negative input by positive 2 keeps it negative.
Decimal multiplication. 83 × 0.5 = 41.5 because half a second adds half of 83 meters.
Domain notation. t ≥ 0 and [0, ∞) include zero and every larger real time while the model applies.
Rational and irrational inputs. 0.5 and $square root of 2$ can both be positive elapsed times; a continuous measurement need not be a whole count.

Why it works. After t seconds, traveling 83 meters each second adds 83t meters. Adding the 250 meters already present gives D(t) = 83t + 250. Each extra second adds another 83, so the plotted points keep the same slant and lie on a straight line. The story also limits which inputs make sense. Here t measures elapsed time after the clock starts, so negative t belongs outside this model even though the formula can calculate it.

RuleRule: A linear function has form f(x) = mx + b, where fixed real numbers m and b give the constant rate m and input-zero output b. Its graph lies on a straight nonvertical line. If m ≠ 0, it is a polynomial of degree 1; if m = 0, it is a constant function.

The same idea, five ways

Say it

Say: output equals a steady amount per step times the input, plus the start.

Write it

A linear function changes output at a constant rate as input changes.

In math

f(x) = mx + b
y = mx + b
D(t) = 83t + 250
m ≠ 0: degree 1
m = 0: constant function
b = f(0); m and b are fixed real numbers

A moving walkway covers the same extra distance during equal amounts of time.

See it

The input runs horizontally and the output runs vertically.

The same idea, other ways

As a story

Start with what you already have, then add the same amount each step. The train already 250 meters away adds 83 meters every second.

The machine combines starting distance and distance traveled.

With small numbers

In f(x) = 2x + 1, each extra input step adds two. The starting one is present once.

Each input step of one adds two in the output row.

As a picture

A straight ramp keeps the same slant wherever you stand. A linear graph keeps the same change for equal horizontal steps.

The input runs horizontally and the output runs vertically.

Why the change stays steady

Changing x to x + 1 changes mx + b to m(x + 1) + b = mx + m + b. The extra amount is m, while b stays fixed.

f(x) = mx + b

f(x + 1) = mx + m + b

One input step adds m

One extra input step contributes one more copy of m.

.1Words

You can describe a rule before you write any symbols. Imagine you have one dollar saved and put away two more dollars each week. Your total is your original dollar plus two dollars for each week. That sentence says both where you begin and how you change. It is the word form of a linear function. The train sentence works the same way. You begin measuring when the train is 250 meters away, then each second adds 83 meters to that distance.

Rule: Word form names the input, output, starting amount, and constant rate, because those pieces identify what the function means.
Rule: A constant rate of change means equal input changes produce equal output changes, because the same amount is added for each input unit.

Input: seconds after the clock starts

Output: meters from the station

Start: 250 meters

Change: 83 meters per second

The word form names the roles of all four pieces.

Reminder

Repeated addition. Two equal groups of 83 are 83 + 83 = 2 × 83 = 166.

The same idea, five ways

Say it

Say: start with one, then add two for each step.

Write it

Total savings equal one dollar already saved plus two dollars for each week.

In math

f(x) = 2x + 1
D(t) = 83t + 250
output = rate × input + start

A jar begins with money inside, and you add the same deposit each week.

See it

The words identify the distance at the start and the distance added each second.

Worked exampleSay a saving rule and the train rule

Describe the relationship means say the starting amount and the change per step. First describe one dollar saved plus two dollars per week. Then describe the train starting 250 meters from the station and moving away at 83 meters per second.

The two-second train story becomes one pass through the distance machine.

After two weeks, savings are 1 + 2 + 2 = 5 dollars.The starting dollar is counted once, while the deposit is counted once per week.
Name train time in seconds as the input and distance from the station in meters as the output.Distance depends on how much time has passed.
Say: the train is 250 meters away at the start, plus another 83 meters away for each second.The 250 is already present when the clock starts.
For two seconds, calculate 250 + 83 + 83 = 416 meters.Two seconds contribute two equal one-second distances.

Answer

Savings: one dollar at the start plus two dollars per week.
After two weeks, five dollars.
Train: 250 meters at the start plus 83 meters per second.
After two seconds, 416 meters.

Check The multiplication versions agree: 1 + 2 × 2 = 5 and 250 + 83 × 2 = 416.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: The train starts at zero meters because the clock starts at zero seconds.

A zero input does not require a zero output. The clock starts after the train is already 250 meters away.

✓ Instead: At zero seconds in this model, the distance is 250 meters.

Tips and tricks

Tip: Include both 'at the start' and 'for each' when you describe the rule.

.2Function notation

A function name is a label on a machine. In f(x), the letter f names the machine and the parentheses name its input. It does not mean f times x. For the train, D names the distance machine and t names the seconds you put in. The formula D(t) = 83t + 250 says to multiply seconds by 83 and add 250. This is slope-intercept form: the number multiplying the input is the steady change, and the added number is the output at input zero. The coefficient is the multiplying number, and the exponent is the raised number counting factors: $x to the power 1$ = x.

A polynomial is a sum of number multiples of nonnegative whole-number powers of a variable. Its degree is the largest exponent with a nonzero coefficient, so 2 $x to the power 1$ + 1 has degree 1.
Rule: f(x) names the output at input x, because the parentheses specify what enters the function.
Rule: Slope-intercept form is f(x) = mx + b or y = mx + b, where m is the slope and b is the output at zero.
Rule: mx + b has degree 1 when m ≠ 0, because $x to the power 1$ is its highest power with a nonzero coefficient. When m = 0, a nonzero constant has degree 0; the zero polynomial has no ordinary degree.

D(t) = 83t + 250

t: input in seconds

83: constant rate

250: output at input zero

The formula keeps the input, rate, and starting amount in separate roles.

Reminder

Order of operations. Multiply before adding: 2 × 3 + 1 = 6 + 1 = 7.

The same idea, five ways

Say it

Say: D of t equals eighty-three times t plus two hundred fifty.

Write it

The distance at time t equals eighty-three times elapsed seconds plus 250 meters.

In math

D(t) = 83t + 250
f(x) = mx + b
y = mx + b
$x to the power 1$ = x
m ≠ 0: degree 1
m = 0: f(x) = b
Polynomial: 2 $x to the power 1$ + 1
Degree: highest remaining exponent, here 1
Coefficient: multiplying number, here 2

A labeled recipe tells you what to do to any amount you put in.

See it

The notation names the machine, input, and output.

Worked exampleReplace the input in two formulas

Evaluate means find the output for the named input. Find f(1) for f(x) = 2x + 1, then D(0.5) for the train formula D(t) = 83t + 250.

Elapsed time can be a fraction of a second.

f(1) = 2 × 1 + 1 = 3.The input 1 replaces x, and multiplication happens before addition.
D(0.5) = 83 × 0.5 + 250.The input is half a second, so 0.5 replaces t.
83 × 0.5 = 41.5, then 41.5 + 250 = 291.5 meters.Half a second adds half of one second's 83 meters.

Answer

f(1) = 3.
D(0.5) = 291.5 meters.

Check Half the one-second added distance is 83 ÷ 2 = 41.5 meters. Adding it to 250 gives 291.5, matching substitution.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: D(2) means multiply D by 2.

D is a function name, and the parentheses name its input.

✓ Instead: D(2) = 83 × 2 + 250 = 416 meters.

✗ Not this: Counterexample: Every mx + b has degree 1 even when m = 0.

A zero coefficient removes the input term, so the highest remaining power is no longer .

✓ Instead: 3x + 2 has degree 1; 0x + 7 = 7 is a degree-zero constant.

Tips and tricks

Tip: Say 'of' when reading parentheses: D(2) is 'D of two.'

.3Table

A table is like a receipt that keeps matching items in the same column. Here the item in the top row is an input, and the item directly below it is the matching output. This is tabular form. To make a table from a formula, you choose an allowed input, put it into the formula, and place its answer below it. The train's table shows selected times and distances. It does not show every possible time. You can still use the formula between the pictured columns.

Rule: A table column pairs an input with its output, because the two entries record one use of the function.
Rule: Equal input gaps in a linear function give equal output gaps, because its rate stays constant. Unequal input gaps need proportionally scaled changes.

Each train time is paired with its distance in the same column.

Reminder

Subtraction as change. Later minus earlier gives 499 − 416 = 83.

The same idea, five ways

Say it

Say: this input has the output directly below it.

Write it

A function table displays selected matching inputs and outputs.

In math

table column gives (x, f(x))
D(2) = 416 gives (2, 416)
D(t) = 83t + 250

A receipt keeps each item's name and price together.

See it

Each train time is paired with its distance in the same column.

Worked exampleRead a small table and the train table

Read the matching output means look directly below the requested input. First use the small table to find f(1). Then use the train table to find D(3) and verify its last one-second change.

Each input step of one adds two in the output row.

Each train time is paired with its distance in the same column.

Read the entry directly below 1 in the small table: f(1) = 3.Each column keeps one input with its matching output.
Read the entry directly below 3 in the train table: D(3) = 499 meters.The upper entry gives time and the lower entry gives distance.
Compare columns under 2 and 3: time changes by 3 − 2 = 1 second and distance by 499 − 416 = 83 meters.Later minus earlier measures the change.
Calculate D(3) = 83 × 3 + 250 = 249 + 250 = 499 meters.The formula independently verifies the table answer.

Answer

Small table: f(1) = 3.
Train table: D(3) = 499 meters.
The last one-second change is 83 meters.

Check Three equal additions give 250 + 83 + 83 + 83 = 499 meters too.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: For D(2), read the output in the neighboring column under 3.

That switches which input the output belongs to.

✓ Instead: Stay under 2 and read 416 meters.

✗ Not this: Counterexample: Only the whole seconds in the table are allowed.

The table samples the function; it does not define all possible times.

✓ Instead: A half second is allowed too, giving D(0.5) = 291.5 meters.

Tips and tricks

Tip: Place a finger on the input and move straight down.

.4Graph and allowed inputs

A graph is a map of the relationship. Move sideways to an input, then upward or downward to its output. The pair becomes a point on the picture. This is graphical form. The train's points line up, but you keep only zero seconds and later. The domain means the allowed inputs. Nonnegative means zero or more. Time can include part of a second, so this domain contains every nonnegative real number. A real number is a position on the number line, including whole numbers, fractions, and irrational numbers such as $square root of 2$ .

Rule: A graph plots allowed pairs (input, output), because horizontal coordinates give inputs and vertical coordinates give outputs.
Rule: The train domain is t ≥ 0, or [0, ∞), while the stated speed lasts, because t is elapsed time after the clock starts.
Rule: The unrestricted formula f(x) = 2x + 1 has domain (−∞, ∞), because doubling and adding one work for every real input.
Rule: A continuous-input context can restrict a linear graph to a ray, which starts at one endpoint and continues in one direction, or a segment, which has two endpoints, because only the allowed portion describes that context.

The input runs horizontally and the output runs vertically.

Reminder

Interval endpoints. [0, ∞) includes zero. Infinity uses a parenthesis because it is not an endpoint you reach.

The same idea, five ways

Say it

Say: t is zero or more; the unrestricted x can be any real number.

Write it

The train accepts nonnegative elapsed times while the unrestricted formula accepts all real inputs.

In math

t ≥ 0
[0, ∞)
{t | t ≥ 0}
f(x) = 2x + 1: x is real
(−∞, ∞)
{x | x is real}
Ray domain: t ≥ 0; segment domain: 0 ≤ t ≤ 4

A map can show a road in both directions while your trip begins at a chosen starting point.

See it

The train domain includes zero and all greater real times while the speed stays constant.

Worked exampleA negative algebra input and a half-second train input

Check an input means decide whether its meaning is allowed before calculating. Evaluate f(−1) for unrestricted f(x) = 2x + 1. Then decide whether −1 and 0.5 seconds belong to the train model, and plot the allowed half-second point.

The input runs horizontally and the output runs vertically.

f(−1) = 2 × (−1) + 1 = −2 + 1 = −1.The unrestricted formula accepts every real input, including −1.
Exclude t = −1 from the elapsed-time train model.It lies before the model's chosen starting time.
Keep t = 0.5, then calculate D(0.5) = 83 × 0.5 + 250 = 291.5 meters.Half a second is nonnegative, and time is not limited to whole seconds.
Plot (0.5, 291.5) and keep t ≥ 0.The point puts seconds first and meters second, and the graph must follow the domain.

Answer

Unrestricted formula: f(−1) = −1.
Train: −1 second is excluded.
0.5 second is allowed.
Train point: (0.5, 291.5), with domain [0, ∞) while the stated speed lasts.

Check The half-second point is 41.5 meters beyond 250. Doubling 41.5 gives 83, exactly one second's added travel.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: D(−1) = 167 means negative time is allowed in this model.

A calculable output does not make the input meaningful in the story.

✓ Instead: Use t ≥ 0, because time is measured after the selected start.

Tips and tricks

Tip: Write what the input means beside its domain restriction.
Tip: A ray has one endpoint and continues; a segment has two endpoints. A finite graph window can display only part of either one.

Strategy: step by step

1. Name the input and output with their units.
2. Find the starting amount b and constant rate m.
3. Write words and formula, calculate table columns, and plot the same pairs.
4. Keep the inputs permitted by the story.

Strategy

Strategy: Show one steady relationship four ways

Does the input describe a context with limits?

YesUse the story's limits, such as t ≥ 0 for elapsed time.

NoAn unrestricted mx + b permits every real input.

↓

Is m zero?

YesThe function is constant and the line horizontal.

NoThe polynomial has degree 1 and the line slopes.

1. State the start and constant amount per step.
2. Write output = rate × input + start.
3. Place calculated pairs in an input-output table and plot matching points.
4. Limit the graph to the allowed domain.

Worked exampleA small rule, then the train

Find an output means put in the named input and calculate what comes out. Evaluate f(1) for f(x) = 2x + 1. Then represent D(t) = 83t + 250 in words, formula, table, and graph at two seconds.

Each train time is paired with its distance in the same column.

The input runs horizontally and the output runs vertically.

f(1) = 2 × 1 + 1 = 3.The input replaces x, then multiplication happens before addition.
Say: the train starts 250 meters away and adds 83 meters each second.The starting amount and the per-second amount describe different parts of the trip.
D(2) = 83 × 2 + 250 = 166 + 250 = 416 meters.Two seconds add two groups of 83 to the starting distance.
Read 416 directly below 2 in the train table.Each table column pairs its input with its output.
Plot (2, 416), include (0, 250), and retain t ≥ 0.Input is horizontal and output is vertical; elapsed time begins at zero.

Answer

Small rule: f(1) = 3.
Train words: 250 meters at the start plus 83 meters per second.
Train formula: D(t) = 83t + 250, t ≥ 0 while the speed lasts.
At two seconds: 416 meters, shown by the table column and point (2, 416).

Check Two equal one-second additions give 250 + 83 + 83 = 416, agreeing with formula, table, and graph.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: Every function that increases is linear, including for x ≥ 0.

The square function gains one from zero to one, then three from one to two. Its rate is not constant.

✓ Instead: A linear function gives equal output changes for equal input changes, as 2x + 1 does.

✗ Not this: Counterexample: The train's formula accepts all real inputs, so its model must too.

The story defines elapsed time after a start, excluding negative time.

✓ Instead: Use t ≥ 0 while the stated constant-speed motion lasts.

✗ Not this: Counterexample: A constant function cannot belong with linear functions because it is not degree 1.

This section includes horizontal lines in f(x) = mx + b. The degree-one description applies when m ≠ 0.

✓ Instead: f(x) = 7 is a constant member with slope zero; nonzero m gives degree 1.

Tips and tricks

Tip: Remember 'start plus steady change' for mx + b.
Tip: Words, formula, table, and graph should agree at every pictured input.

Trap. Tip: Count the starting 250 once. Writing 333t would repeat the starting distance at every second.