Fungsi Linear dan Eksponensial
Diperkenalkan| English | Makna |
|---|---|
| linear function/ˈlɪnɪə ˈfʌŋkʃn/ | fungsi linear |
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | fungsi eksponensial |
| proportional/prəˈpɔːʃənl/ | berbanding lurus |
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.
Pada interval input yang sama, fungsi linear berubah dengan…
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Garis memiliki kemiringan konstan, sehingga langkah sama dalam $x$ menambahkan jumlah yang sama ke $y$ — seperti barisan aritmatika.
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.
Saksikan fungsi eksponensial melampaui garis lurus apa pun
y = a·bˣ
Angkat basis b di atas 1 dan kurva akan mengalikan setiap langkah, segera meninggalkan garis mana pun di belakang. Atur b di bawah 1 untuk mendapatkan peluruhan.
Pada interval input yang sama, fungsi eksponensial berubah dengan…
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Eksponensial mengalikan dengan faktor yang sama pada setiap langkah yang setara — versi kontinu dari barisan geometri.
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.
Perubahan eksponensial adalah ____ perubahan: kenaikan merupakan persentase tetap dari jumlah saat ini.
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Proportional perubahan berarti setiap langkah diskalakan berdasarkan nilai saat ini — tumbuh 10% dari apa pun yang Anda miliki sekarang.
Populasi yang berlipat ganda setiap tahun paling baik dimodelkan oleh fungsi eksponensial.
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Berlipat ganda adalah mengalikan dengan 2 setiap tahun — faktor konstan pada interval yang setara, ciri khas pertumbuhan eksponensial.
Pilih semua pernyataan yang benar.
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Menambah jumlah yang sama adalah linear, bukan eksponensial. Tiga lainnya benar dalam memasangkan ide-ide tersebut.
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.