Exponentials and logarithms (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Exponentials and logarithms (AS Unit 1: Pure Mathematics A) explained
The exponential function f(x) = aˣ with base a > 0 is defined for all real x, with range f(x) > 0.
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When a > 1 the function shows exponential growth: it is strictly increasing, passes through the y-intercept (0, 1) and the point (1, a). As x → −∞, aˣ approaches zero from above, so the line y = 0 is a horizontal asymptote; as x → +∞, aˣ increases without bound. When 0 < a < 1 the function shows exponential decay: strictly decreasing, passing through (0, 1), with horizontal asymptote y = 0 as x → +∞. The graph never touches or crosses the x-axis. In applications, aˣ models unconstrained multiplication such as doubling populations (2ˣ). Transformations behave as expected: y = aˣ + k shifts the horizontal asymptote to y = k, and y = a^(x − h) shifts the graph horizontally.
Your focus
- Sketch exponential functions y = aˣ for a > 1 and 0 < a < 1 with key features labelled.
- Identify the domain (ℝ), range (y > 0), and horizontal asymptote (y = 0) of exponential curves.
- Apply vertical and horizontal transformations to exponential functions.
Exponentials and logarithms (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- Sketching an exponential curve showing the correct growth or decay orientation according to whether a > 1 or 0 < a < 1.
- Marking the y-intercept at (0, 1) and identifying the point (1, a) on the curve.
- Drawing and labelling the horizontal asymptote along y = 0, showing the curve approaching but never touching or crossing it.
- Stating the domain (x ∈ ℝ) and range (f(x) > 0) of y = aˣ.
- Applying vertical and horizontal transformations, e.g. y = 2ˣ + 3 has horizontal asymptote y = 3.
Examiner Tips
- 💡Draw the curve getting progressively closer to the x-axis without ever touching or crossing it.
- 💡Label the y-intercept (0, 1) clearly on every sketch of an untransformed exponential function, and mark the asymptote y = 0.
Common Mistakes
- Allowing the curve to cross the x-axis or curl away from the horizontal asymptote y = 0; the curve approaches the axis asymptotically.
- Drawing exponential decay curves with a vertical asymptote instead of approaching y = 0 as x → +∞.
- Forgetting that the y-intercept is always (0, 1) for any base a > 0, since a⁰ = 1.