Linear and Exponential Functions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ | hàm bậc nhất |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | hàm mũ |
| proportional/prəˈpɔːʃənl/ | tỷ lệ |
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…
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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…
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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.
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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 a linear function.
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False. Doubling means multiplying by 2 each year, a constant factor over equal intervals, which is exponential. Linear models add a constant amount.
Select all true statements.
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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.
Write from a fifth term of 28
- An arithmetic sequence starts at 4 and adds 6 each time.
- Find the fifth term.
An arithmetic sequence has first term 4 and common difference 6. What is the fifth term?
4 plus 24 is 28.
What is the fifth term?
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Four additions of 6 reach 28. Other options — 10 (the second term); 6 (the common difference).