Solutions — Exp/Log 13C Worksheet

Year 11 Methods Unit 1 · Cambridge §13C · Mr Wong

ANSWER KEY

Warm-up — tables of values

Q1. Filled tables:

Function$x=-2$$x=-1$$x=0$$x=1$$x=2$
$y=2^x$$\tfrac{1}{4}$$\tfrac{1}{2}$$1$$2$$4$
$y=5^x$$\tfrac{1}{25}$$\tfrac{1}{5}$$1$$5$$25$
$y=\left(\tfrac{1}{2}\right)^x$$4$$2$$1$$\tfrac{1}{2}$$\tfrac{1}{4}$
$y=3^x$$\tfrac{1}{9}$$\tfrac{1}{3}$$1$$3$$9$

Q2. Untransformed $y=a^x$ (any allowed base):

· Domain: $\mathbb{R}$
· Range: $\mathbb{R}^+$ i.e. $(0,\infty)$
· Asymptote: $y=0$
· $y$-int: $(0,1)$

Part A — Asymptote, $y$-intercept, range

Q3. Add the constant for the asymptote, evaluate at $x=0$ for the $y$-intercept.

a $y=2^x+5$  →  asy $y=5$, $y$-int $(0,6)$, range $(5,\infty)$
b $y=2^x-3$  →  asy $y=-3$, $y$-int $(0,-2)$, range $(-3,\infty)$
c $y=3\times 2^x$  →  asy $y=0$, $y$-int $(0,3)$, range $(0,\infty)$
d $y=-2^x+1$  →  asy $y=1$, $y$-int $(0,0)$ (since $-2^0+1=0$), range $(-\infty,1)$
e $y=4\cdot 5^{-x}-2$  →  asy $y=-2$, $y$-int $(0,2)$, range $(-2,\infty)$
f $y=-3\cdot 2^{-4x}+5$  →  asy $y=5$, $y$-int $(0,2)$, range $(-\infty,5)$

Part B — Sketches (key features only)

Q4–6. Asymptote (dashed), $y$-intercept, monotonic shape.

4 $y=2^x+3$: asy $y=3$, $y$-int $(0,4)$, range $(3,\infty)$; increasing, no $x$-intercept.
5 $y=2\times 3^{2x}+1$: asy $y=1$, $y$-int $(0,3)$, range $(1,\infty)$; increasing steeply, no $x$-intercept.
6 $y=-3^{2x}+4$: asy $y=4$, $y$-int $(0,3)$, range $(-\infty,4)$; decreasing through the $x$-axis. CAS gives $x$-int $\approx 0.631$ (since $3^{2x}=4 \Rightarrow x=\tfrac{\ln 4}{2\ln 3}$).

Part C — Identify and apply

Q7. Matching table:

FunctionAsymptote$y$-interceptRange
$y=3^x-4$$y=-4$$(0,-3)$$(-4,\infty)$
$y=5\times 2^x$$y=0$$(0,5)$$(0,\infty)$
$y=-4^x+6$$y=6$$(0,5)$$(-\infty,6)$
$y=2\times 3^{-x}+1$$y=1$$(0,3)$$(1,\infty)$

Q8. $P=4\times 2^{0.5\,t}$ (thousands):

a $t=0$: $P=4\times 2^0=$ $4000$ people
b $t=4$: $P=4\times 2^{2}=4\times 4=$ $16\,000$ people
c No upper bound — exponential growth with no horizontal asymptote above it; as $t\to\infty$, $P\to\infty$.
Y11 Methods · Exp/Log 13C · Solutions Page 1 of 1 · Hand back to Mr Wong