Linear and Exponential Functions
Introduced| English |
|---|
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ |
| proportional/prəˈpɔːʃənl/ |
Add versus multiply, again
- Two savings plans: one adds $\$100$ a year, the other grows by $10\%$ a year.
- At first the flat $\$100$ looks better, but the percentage plan quietly pulls ahead.
- These are the two great patterns of change: linear and exponential.
- They are the continuous cousins of arithmetic and geometric sequences.
Arithmetic pairs with linear
- An arithmetic sequence adds a constant; a linear function 线性函数 does the same for every input.
- Over each equal step in $x$, a line adds the same amount to $y$ — its constant slope.
- Plot the arithmetic terms and they land right on a straight line.
- Steady addition is the fingerprint of linear change.
Over equal input intervals, a linear function changes by…
A line has a constant slope, so equal steps in $x$ add the same amount to $y$ — like an arithmetic sequence.
Geometric pairs with exponential
- A geometric sequence multiplies by a constant; an exponential function 指数函数 does the same continuously.
- Over each equal step in $x$, an exponential multiplies $y$ by the same factor.
- So the outputs form a geometric sequence: $a, ab, ab^2, ab^3, \dots$
- Steady multiplication is the fingerprint of exponential change.
Watch an exponential outrun any straight line
y = a·bˣ
Raise the base b above 1 and the curve multiplies each step, soon leaving any line behind. Set b below 1 to get decay.
Over equal input intervals, an exponential function changes by…
An exponential multiplies by the same factor each equal step — the continuous version of a geometric sequence.
Proportional change
- Exponential growth is proportional 成比例 change: each increase is a percentage of the current amount.
- Grow $10\%$ a year and the yearly gain gets bigger as the total gets bigger.
- Linear growth adds the same absolute amount regardless of the current size.
- "Percentage per period" always signals an exponential, never a line.
Exponential change is ____ change: the increase is a fixed percentage of the current amount.
Proportional change means each step scales with the current value — grow 10% of whatever you have now.
A population that doubles every year is best modeled by an exponential function.
Doubling is multiplying by 2 each year — a constant factor over equal intervals, the signature of exponential growth.
Select all true statements.
Adding equal amounts is linear, not exponential. The other three correctly pair the ideas.
Spotting which model
- Look at how the data changes over equal steps: constant difference → linear; constant ratio → exponential.
- A table helps: subtract to test for linear, divide to test for exponential.
- Over a long enough range, exponential growth beats any linear function, no matter how steep.
- Match the model to the pattern of change before you fit numbers.
Do not judge by the first few values. Early on a line can sit above an exponential, so a quick glance misleads. Test the pattern of change — same difference or same ratio — not just the current heights.
A pond's lily pads: $10, 20, 40, 80, \dots$ each day.
- Each day the count is multiplied by $2$ — a constant ratio.
- Constant ratio → an exponential function, not a line.
- Rule: pads $= 10 \cdot 2^{\,d}$, doubling without limit until the pond fills.
An arithmetic sequence pairs with a linear function (adds a constant); a geometric sequence pairs with an exponential function (multiplies by a constant). Exponential growth is proportional — a percentage of the current amount — and it eventually overtakes any line.