Writing & Transforming Linear Functions

Once a linear model is written, every question about it — rate, starting value, prediction, shift — is a short calculation.

Your phone bill arrives. It has two lines on it.

One says monthly charge, $25.00. The other says 340 texts at $0.10, $34.00.

read the bill, then write the rule PHONE BILL monthly charge $25.00 340 texts @ $0.10 $34.00 total $59.00 send no texts at all and you still owe $25 C = 0.10t + 25 the fixed line on the bill is b; the per-text line is m

Notice how differently those two behave. Send no texts at all next month and you still owe the $25.

The $34 exists only because of the texts, and it would grow or shrink with them.

So the bill splits into a part that happens once and a part that happens per text.

Two questions turn any such story into an equation. What is true at zero? — that is b. What happens per one more? — that is m. Here: C = 0.10t + 25.

Changing b slides a line; m keeps it parallel change b and the line slides; change m and it tilts y = x + 2 y = x y = x - 3 same m, so all three stay parallel f(x) + k moves the graph up k; f(x + k) moves it left k, which surprises nearly everyone

Writing a model from a description

Find the fixed part: what is true at the very start, before anything happens. That is b.

Find the per-unit part: what changes for each additional one. That is m.

"$30 to join plus $12 a month" is C(t) = 12t + 30. Say what t means in words as well.

Two questions, in order

What is the value at zero? What happens per step? Answer those two and the equation is written.

Average rate of change

For a linear function the rate of change is the same everywhere, so the average rate equals the slope.

For any other function it varies, and the formula f(b) - f(a)b - a gives the slope of the straight line joining the two points.

It answers "on average, how fast did it change between here and there?"

It is just a slope

The average rate of change over [a, b] is the slope between (a, f(a)) and (b, f(b)) — nothing more exotic than rise over run.

Transformations

f(x) + k shifts the graph up k units. f(x) - k shifts it down.

f(x + k) shifts it left k units; f(x - k) shifts it right. This is the one that surprises everybody.

-f(x) reflects it across the x-axis. af(x) stretches it vertically by a factor of a.

Inside the brackets is backwards

Changes to x act in the opposite direction to what the sign suggests. f(x - 3) moves right. Think: to get the old output, x now has to be 3 bigger.

Arithmetic sequences

A sequence with a constant difference is a linear function on the whole numbers.

Explicit rule: an = a1 + (n - 1)d, where d is the common difference.

Recursive rule: a1 = start and an = an-1 + d. Both describe the same list.

Why (n - 1)?

You take the step d one fewer time than the term number, because the first term needs no steps at all.

Worked examples

Writing a model

A phone plan costs $25 a month plus $0.10 per text. Write C(t) and find the cost of 340 texts.

  1. At zero texts the cost is $25, so b = 25. The fixed part.
  2. Each text adds $0.10, so m = 0.10. The per-unit part.
  3. C(t) = 0.10t + 25 With t the number of texts.
  4. C(340) = 0.10(340) + 25 = 34 + 25 = $59 Substitute and evaluate.

C(t) = 0.10t + 25; $59

Average rate of change

For f(x) = x2 + 1, find the average rate of change from x = 1 to x = 4.

  1. f(1) = 1 + 1 = 2 Evaluate at the left end.
  2. f(4) = 16 + 1 = 17 And at the right end.
  3. 17 - 24 - 1 = 153 = 5 Change in output over change in input.
  4. So on average it rose 5 per unit over that stretch. Not constant here — this is an average.

5

Arithmetic sequence

A sequence starts 7, 11, 15, 19, …. Find the 30th term.

  1. Common difference: 11 - 7 = 4 Check another pair: 15 - 11 = 4 ✓
  2. an = 7 + (n - 1)(4) First term plus (n-1) steps.
  3. a30 = 7 + 29(4) = 7 + 116 29 steps, not 30.
  4. a30 = 123 Equivalently 4n + 3, and 4(30) + 3 = 123 ✓

123

Words to know

Linear model
An equation of the form f(x) = mx + b fitted to a situation.
Rate of change
The slope, in the units of the situation.
Average rate of change
f(b) - f(a)b - a over an interval.
Transformation
A change to the function that moves or stretches its graph.
Arithmetic sequence
A list with a constant difference. Its rule is linear.

Mistakes to watch for

Swapping m and b in a word problem. The fixed, one-off amount is b; the repeated per-unit amount is m. Ask "what do I owe before anything happens?" and you have b directly.

Shifting the wrong way for f(x - 3). Changes inside the brackets act in reverse: subtracting moves the graph right. Try it with one point — f(0) now happens when x = 3.

Using n where the formula needs n - 1. The tenth term takes nine steps, because the first term needed none. This single off-by-one causes more sequence errors than everything else combined.

Reporting an average rate of change with no units. It is a rate — "5 points per hour," not "5." The bare number cannot be checked for sense, and a sense check is what catches the sign errors.

The idea that a graph could be shifted by changing the equation is surprisingly recent as a teaching tool, but the surprise inside it is ancient.

f(x - 3) shifts the graph right, not left — because to get the output the old graph gave at x, the new one needs an input 3 larger.

Nearly every student meets this once and disbelieves it. Substituting a single number settles it faster than any argument.

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