This introduction covers The Power of Doubling within Exponential Graphs for GCSE Mathematics. Revise Exponential Graphs in Graphs for GCSE Mathematics with 11 exam-style questions and 10 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 1 of 10 in this topic. Use this introduction to connect the idea to the wider topic before moving on to questions and flashcards.
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 — doubling each time. The king agreed, thinking it a small reward. By square 64, the total would be 18 billion billion grains — 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 — all from the deceptively simple equation y = aˣ.
Practice questions for Exponential Graphs
The graph of y = 3ˣ always passes through which point?
Explain why the graph of y = 3ˣ has a horizontal asymptote at y = 0, and state the domain of values that y can take.
Quick recall flashcards
What is the asymptote of y = aˣ?
The x-axis (the line y = 0) is a horizontal asymptote.
For growth (a > 1): as x → -∞, y → 0 but never reaches 0
For decay (0 < a < 1): as x → +∞, y → 0 but never reaches 0
The graph gets infinitely close to the x-axis but never crosses it.
y is always positive — it never equals zero.
What is the y-intercept of any graph y = aˣ?
The y-intercept is always (0, 1).
Reason: when x = 0, y = a⁰ = 1 for any base a.
This is true for y = 2ˣ, y = 3ˣ, y = 5ˣ, and even y = (0.5)ˣ.
All exponential graphs of the form y = aˣ pass through (0, 1).