Recognise exponential growth and decay

By UK GCSE Mathematics · independent study · myAudience

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Recognise exponential growth and decay

In an exponential rule y = kˣ, the variable is the exponent, not a factor multiplying k. For k > 1 the graph grows as x increases; for 0 < k < 1 it decreases. If k = 1, every output is 1, so the graph is constant.

For every positive base, k⁰ = 1. The graph therefore passes through (0, 1), and its outputs remain positive. Growth by equal multiplicative factors over equal input intervals distinguishes an exponential from a linear rule.

For y = 2ˣ, inputs −1, 0, 1, 2 give 1/2, 1, 2, 4. Its gradient changes, and the curve approaches the horizontal axis on the left without reaching zero. A sketch should preserve that behaviour rather than join points with one straight line.

Worked example: Find the outputs of y = (1/2)ˣ for x = 0, 1, 2.

Apply the power to the entire fraction.

Compare successive outputs.

Answer: 1, 1/2, 1/4; each step halves the output.

Check your understanding: Does y = 2x mean the same thing as y = 2ˣ?

Answer: No.

Why: One is linear multiplication; the other has a variable exponent.