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The Power of Doubling
The Power of Doubling
An ancient legend tells of a chess player who asked a king for one grain of rice on the first square, two on the second, four on the third โ doubling each time. The king agreed, thinking it a small reward. By square 64, the total would be 18 billion billion grains โ more than all the rice ever grown. That is exponential growth: slow at first, then exploding. It describes population growth, compound interest, radioactive decay, and the spread of diseases โ all from the deceptively simple equation y = aหฃ.
What is the asymptote of y = aหฃ?: The x-axis (the line y = 0) is a horizontal asymptote.
For growth (a > 1): as x โ -โ, y โ 0 but never reaches 0
For decay (0 < a < 1): as x โ +โ, y โ 0 but never reaches 0
The graph gets infinitely close to the x-axis but never crosses it.
y is always positive โ it never equals zero.
Mathematics glossary
- What is the asymptote of y = aหฃ?
- The x-axis (the line y = 0) is a horizontal asymptote. For growth (a > 1): as x โ -โ, y โ 0 but never reaches 0 For decay (0 < a < 1): as x โ +โ, y โ 0 but never reaches 0 The graph gets infinitely close to the x-axis but never crosses it. y is always positive โ it never equals zero.
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๐ก Exam Tips for Exponential Graphs
- y-intercept is always (0, 1) for y = aหฃ โ mark this point first when sketching
- Curve never crosses x-axis โ draw it approaching but never touching the x-axis
- Growth vs decay: growth curves rise to the right; decay curves fall to the right
- For y = k ร aหฃ: the y-intercept is (0, k), not (0, 1)
- Compound interest: use the formula A = P(1 + r/100)โฟ โ the multiplier is (1 + r/100) for growth, (1 โ r/100) for decay
- Check if exponential: calculate the ratio of consecutive terms โ if constant, it is exponential
Now try it yourself
Quiz ยท Question 1 of 11
The graph of y = 3หฃ always passes through which point?
Tap an answer to check it