Graphs of Log & Exponential Functions

Two base shapes cover this subtopic, and every exam sketch is one of them shifted, stretched or reflected. The marks attach to three features: intercept, asymptote, behaviour.

The base shapes

$y = a^x$ ($a > 1$), including $y = e^x$: through $(0, 1)$; always positive; $x$-axis is the asymptote (left end); climbs steeply right. For $0 < a < 1$ the curve decays instead, $y = \left(\tfrac{1}{2}\right)^x$ is $y = 2^x$ reflected in the $y$-axis.

$y = \ln x$ (and $\log_a x$): through $(1, 0)$; exists only for $x > 0$; $y$-axis is the asymptote; rises slowly forever. $y = \ln x$ is $y = e^x$ reflected in the line $y = x$, the inverse-pair geometry, itself a sketchable mark.

Transformations: track the anchor features

Move the intercept and the asymptote with the transformation, and the sketch draws itself:

• $y = e^x + 3$: asymptote lifts to $y = 3$; intercept $(0, 4)$; range $f(x) > 3$
• $y = e^{x-2}$: shifts right; intercept $(0, e^{-2})$; asymptote still $y = 0$
• $y = -e^x$: reflected in the $x$-axis; everything negative; asymptote $y = 0$ approached from below
• $y = \ln(x - 2)$: shifts right; vertical asymptote $x = 2$; intercept $(3, 0)$; domain $x > 2$

State the asymptote as an equation (“asymptote $y = 3$”), naming it is frequently a separate B mark, and the range follows directly from it.

What sketches get used for

• Counting solutions: “how many roots does $e^x = 3 - x$ have?”, sketch both, count intersections (graphical solving)
• Justifying rejections: the graph of $\ln x$ shows instantly why $\ln(\text{negative})$ can’t happen in log equations
• Modelling setups: growth/decay curves $P = P_0 e^{kt}$ with the initial value as intercept, the sketch anchors which way $k$ points

Common mistakes

• $y = e^x$ drawn through the origin (it passes through $(0, 1)$)
• $\ln x$ drawn for $x \le 0$
• Asymptotes unlabelled, or drawn as reached/crossed
• Transformed intercepts left uncomputed (e.g. $(0, e^{-2})$ skipped)
• Decay ($0 < a < 1$) drawn as growth

Full topic context: Logs & Exponentials notes.