Graphs of Exponential Functions (Ex 13C)

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

Name

Class

Date

Learning intentions.

Sketch $y=a^x$ for $a>1$ and for $0<a<1$, with asymptote, $y$-intercept and range
Apply dilations, translations and reflections to produce $y=k\,a^{nx}+c$ style functions
State the asymptote, $y$-intercept and range for a transformed exponential

Key results — $y=a^x$ ($a>0,\;a\ne 1$)

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

Vertical shift by $c$ moves the asymptote to $y=c$. A leading minus reflects in the $x$-axis.

Warm-up — table of values

1. Complete the tables.

$y=2^x$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$

$y=5^x$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$

$y=\left(\tfrac{1}{2}\right)^x$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$

$y=3^x$

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$

2. For $y=a^x$ (any allowed base), state: domain = range = asymptote: $y=$ $y$-int:

Part A — State the asymptote, $y$-intercept and range

3. For each, fill in the table. Do not sketch yet.

a $y=2^x+5$: asy ____ · y-int ____ · range ____

b $y=2^x-3$: asy ____ · y-int ____ · range ____

c $y=3\times 2^x$: asy ____ · y-int ____ · range ____

d $y=-2^x+1$: asy ____ · y-int ____ · range ____

e $y=4\times 5^{-x}-2$: asy ____ · y-int ____ · range ____

f $y=-3\times 2^{-4x}+5$: asy ____ · y-int ____ · range ____

Exp Graphs 13C — continued

Part B (sketching) · Application

Name

Part B — Sketch on the axes

Sketch each graph on the grid. Label the asymptote (dashed line) and the $y$-intercept.

4. $y=2^x+3$

Asymptote: $y=$ ____ · $y$-intercept: ____ · Range: ____

5. $y=2\times 3^{2x}+1$

Asymptote: $y=$ ____ · $y$-intercept: ____ · Range: ____

6. $y=-3^{2x}+4$ (use CAS for any $x$-intercept)

Asymptote: $y=$ ____ · $y$-intercept: ____ · Range: ____

Part C — Identify and apply

7. Match each rule to its asymptote and $y$-intercept (one row per function).

Function	Asymptote	$y$-intercept	Range
$y=3^x-4$
$y=5\times 2^x$
$y=-4^x+6$
$y=2\times 3^{-x}+1$

8. A population $P$ (in thousands) is modelled by $P=4\times 2^{0.5\,t}$ where $t$ is years from now. (a) What is the current population? (b) What is the population after $4$ years? (c) Does this model have an upper bound? Briefly explain.