Today's lesson
We're moving into the graphical side of Chapter 13 — Exponentials & Logarithms. By the end of today you should be able to:
Learning intentions
- Recognise the shape of $y=a^x$ for $a>1$ (growth) and for $0<a<1$ (decay)
- State the asymptote, the $y$-intercept, the domain and the range of any exponential
- Apply dilations ($y=k\cdot a^x$, $y=a^{nx}$) and translations ($y=a^{x+b}$, $y=a^x+c$, including reflections)
- Sketch a transformed exponential and label its asymptote, $y$-intercept and range
Part 1 — The basic shape: $y=a^x$, $a>1$
An exponential function is one in which the variable sits in the exponent: $$y=a^x,\quad a\in\mathbb{R}^+\setminus\{1\}.$$ Build a table of values for $y=2^x$ and $y=5^x$:
$y=2^x$
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $y$ | $\tfrac{1}{4}$ | $\tfrac{1}{2}$ | $1$ | $2$ | $4$ |
$y=5^x$
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $y$ | $\tfrac{1}{25}$ | $\tfrac{1}{5}$ | $1$ | $5$ | $25$ |
📺 Walkthrough: the shape of $y=2^x$, $y=5^x$ and $y=\left(\tfrac{1}{2}\right)^x$, the common point $(0,1)$, and the $x$-axis as an asymptote.
Properties of $y=a^x$ when $a>1$ (growth)
- Domain: $\mathbb{R}$ Range: $\mathbb{R}^+$ (i.e. $y>0$)
- $y$-intercept at $(0,1)$ — because $a^0=1$ for any $a$
- Horizontal asymptote $y=0$: as $x\to -\infty$, $y\to 0^+$
- As $x\to \infty$, $y\to \infty$ (faster for larger base $a$)
Part 2 — When $0<a<1$ (decay)
Compute a table for $y=\left(\tfrac{1}{2}\right)^x$:
| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
|---|---|---|---|---|---|
| $y$ | $4$ | $2$ | $1$ | $\tfrac{1}{2}$ | $\tfrac{1}{4}$ |
Notice $\left(\tfrac{1}{2}\right)^x=2^{-x}$, so this is just $y=2^x$ reflected in the $y$-axis. Same asymptote, same $y$-intercept, but it now decreases as $x$ increases.
Part 3 — Transformations
From $y=a^x$ to a transformed graph
- $y=k\cdot a^x$ — dilation of factor $k$ from the $x$-axis ($y$-intercept becomes $(0,k)$)
- $y=a^{nx}$ — dilation of factor $\tfrac{1}{n}$ from the $y$-axis
- $y=a^{x-b}$ — translation $b$ units to the right ($y=a^{x+b}$ shifts left)
- $y=a^x+c$ — translation $c$ units up — this moves the asymptote to $y=c$
📺 Walkthrough: $y=2^x$ shifted up 3 to give $y=2^x+3$. Asymptote moves to $y=3$, $y$-intercept becomes $(0,4)$, range becomes $(3,\infty)$.
Part 4 — Worked examples
- Start with $y=2^x$ (growth, asymptote $y=0$, $y$-intercept $(0,1)$).
- Add $3$ — shift the entire graph up $3$ units.
- Asymptote: $\boxed{y=3}$
- $y$-intercept: when $x=0$, $y=2^0+3=1+3=4$, so $\boxed{(0,4)}$
- Range: $\boxed{(3,\infty)}$ (the graph stays strictly above the new asymptote)
Now you try: Sketch $y=2^x-3$. Answer: asymptote $y=-3$, $y$-intercept $(0,-2)$, range $(-3,\infty)$.
- Base shape: $y=3^x$ (growth, asymptote $y=0$, $y$-int $(0,1)$).
- $3^{2x}$ is a horizontal dilation by factor $\tfrac{1}{2}$ (graph grows twice as fast).
- $2\times 3^{2x}$ stretches vertically by factor $2$.
- The $+1$ shifts up by $1$ — asymptote moves to $y=1$.
- At $x=0$: $y=2\times 3^0+1=2\times 1+1=3$, so $y$-intercept $\boxed{(0,3)}$.
- Asymptote: $\boxed{y=1}$ Range: $\boxed{(1,\infty)}$
Now you try: Sketch $y=3\times 2^x-1$. Answer: asymptote $y=-1$, $y$-int $(0,2)$, range $(-1,\infty)$.
- Base shape: $y=3^{2x}$ — growth, $y$-int $(0,1)$, asymptote $y=0$.
- The negative sign out the front reflects the graph in the $x$-axis — it now drops down to the right.
- Add $4$ — shifts everything up $4$, so the asymptote becomes $y=4$.
- At $x=0$: $y=-3^0+4=-1+4=3$, so $y$-intercept $\boxed{(0,3)}$.
- Asymptote: $\boxed{y=4}$ Range: $\boxed{(-\infty,4)}$ (the graph is now strictly below the asymptote).
Domain stays $\mathbb{R}$ for every transformed exponential — only the range and asymptote change with a vertical shift.
Practice 4.1 — for each, state asymptote, $y$-intercept and range. (No sketch needed yet.)
- $y=2^x+5$
- $y=2^x-3$
- $y=3\times 2^x$
- $y=-2^x+1$
- $y=4\times 5^{-x}-2$
- $y=-3\times 2^{-4x}+5$
b) asy $y=-3$; $y$-int $(0,-2)$; range $(-3,\infty)$
c) asy $y=0$; $y$-int $(0,3)$; range $(0,\infty)$
d) asy $y=1$; $y$-int $(0,0)$; range $(-\infty,1)$
e) asy $y=-2$; $y$-int $(0,2)$; range $(-2,\infty)$
f) asy $y=5$; $y$-int $(0,2)$; range $(-\infty,5)$
Part 5 — Quick quiz (5 min)
Pick the correct answer for each, then click Mark.
Q1. The $y$-intercept of $y=a^x$ (for any allowed $a$) is:
Q2. The range of $y=2^x$ is:
Q3. The horizontal asymptote of $y=2^x-3$ is:
Q4. The $y$-intercept of $y=2\times 3^x+1$ is:
Q5. The range of $y=-3^x+4$ is:
Working program — Cambridge Ex 13C
After the quiz, open Cambridge Chapter 13 and work through Exercise 13C:
| Question | Foundation | Standard (set work) | Advanced |
|---|---|---|---|
| Q1 — sketch basic $y=a^x$ | 1 (a,c) | 1* | 1* |
| Q2 — vertical translation | 2 (a,c) | 2* | 2* |
| Q3 — reflection / sign change | 3 (a,c) | 3* | 3* |
| Q4 — combined dilation + translation | 4 (a,b) | 4 | 4 all |
| Q5 — CAS-supported sketching | — | 5 | 5 |
| Q6 — extension | — | — | 6 |
If you finish early, head into Ex 13D — Solving exponential equations and inequalities.
Exit ticket — write in your book
- What is the asymptote of every untransformed $y=a^x$ graph?
- What is the $y$-intercept of $y=2^x+5$?
- If we add a negative sign out the front (e.g. $y=-2^x$), what does it do to the graph?