Graphing Exponential Functions

Pre-Calculus · Axiom Academy

LESSON Graphing Exponential Functions How the y-intercept, the horizontal asymptote, and the base shape every curve of the form . 1. Growth: Rising Off the Asymptote Take the base b = 2 , so f(x) = 2^x . Read it left to right: far to the left the curve is almost flat, hugging the line y = 0 but never crossing it — that line is the horizontal asymptote . It passes through the y -intercept (0, 1) , then climbs faster and faster because each step to the right doubles the height. Anchor points: y -intercept and (1, b) Now shrink the base to , so . Each step to the right now halves the height, so the curve falls. It still crosses (0, 1) , but the asymptote y = 0 is now hugged on the right . Since , this graph is exactly the growth curve reflected across the y -axis. The curve decays — it falls left to right toward the asymptote. As the height , so the curve hugs y = 0 on the right side. Any b^0 = 1 , so f(0) = 1 — the curve still crosses (0, 1) . , the mirror of 2^x across the y -axis. At x = 1 the height is , at x = 2 it is — halving each step. Going left it doubles: f(-1) = 2 , f(-2) = 4 . The range is still ; the curve never reaches the axis. 3. The Coefficient a Sets the y -Intercept Multiply by a coefficient: . Because , the y -intercept lands at (0, a) . Sliding a from 1 up to 3 stretches the whole curve vertically and lifts that intercept — yet the asymptote stays put at y = 0 , because multiplying 0 by a is still 0 . Draw the asymptote y = 0 (the x -axis).

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