This topic summary covers Knowledge Organiser: Exponential Graphs 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 10 of 10 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Exponential Graphs
Key Facts
- y = aˣ: y-intercept always (0, 1)
- Asymptote: y = 0 (never crosses x-axis)
- y always positive
- Growth: a > 1; Decay: 0 < a < 1
- y = k × aˣ: y-intercept is (0, k)
Real-Life Formulas
- Compound interest: A = P(1 + r/100)ⁿ
- Depreciation: V = P(1 − r/100)ⁿ
- Growth multiplier: (1 + r/100)
- Decay multiplier: (1 − r/100)
Key Vocabulary
- Exponential growth: multiplies by a fixed ratio > 1 each period
- Exponential decay: multiplies by a ratio between 0 and 1
- Asymptote: line the curve approaches but never reaches
- Base (a): the constant being raised to a power
Common Errors
- Thinking y-intercept is (0, 0) instead of (0, 1)
- Allowing the decay curve to touch the x-axis
- Confusing exponential growth with linear growth
- Using wrong multiplier in compound interest formula
Key Formulas
- Growth: y = a × bˣ where b > 1
- Decay: y = a × bˣ where 0 < b < 1
- y-intercept always at (0, a) — substitute x = 0
- Asymptote: y = 0 (x-axis) — curve approaches but never touches
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 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).
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.