Coaching Centre
Exponential and logarithmic graphs
The graphs of y = a^x and y = log_a x, their asymptotes and key points, and how reflections, translations and dilations change them.
The two basic graphs
$y = a^x$ ($a > 1$)
- Passes through $(0, 1)$ and $(1, a)$.
- Horizontal asymptote $y = 0$.
- Domain: all real $x$. Range: $y > 0$.
- Always increasing.
$y = \log_a x$ ($a > 1$)
- Passes through $(1, 0)$ and $(a, 1)$.
- Vertical asymptote $x = 0$.
- Domain: $x > 0$. Range: all real $y$.
- Always increasing.
The two graphs are reflections of each other in the dashed line y = x, because a power and a log undo each other.
What happens to both graphs as $a$ gets close to $1$?
Transformations
- Reflections: $y = a^{-x}$ reflects $y = a^x$ in the $y$-axis. $y = -\log_a x$ reflects $y = \log_a x$ in the $x$-axis.
- Translations: $y = a^{x - h} + k$ moves the graph right $h$ and up $k$; the asymptote becomes $y = k$. For $y = \log_a (x - h)$ the asymptote becomes $x = h$.
- Dilations: $y = c \times a^x$ stretches vertically by factor $c$, so the $y$-intercept becomes $c$.
Orange: the transformed curve. Dashed: its horizontal asymptote.
- Make $c$ negative. Which way does the graph reflect?
- Set $h = -3$ and $c = 1$, then set $h = 0$ and $c = 8$. Why do they look the same?
Sketch $y = 2^x - 4$, showing intercepts and the asymptote.
State the domain and asymptote of $y = \log_3 (x + 2)$, and find its $x$-intercept.
Every graph $y = a^x$ (with $a > 0$) passes through:
$a^0 = 1$.
$y = \log_3 x$ passes through:
$\log_3 1 = 0$ and $\log_3 3 = 1$.
$y = 2^x + 3$ has horizontal asymptote $y = k$. Find $k$.
Shifting up $3$ moves the asymptote $y = 0$ to $y = 3$.
The domain of $y = \log_2 (x - 1)$ is:
Need $x - 1 > 0$.
$y = 2^{-x}$ is the reflection of $y = 2^x$ in:
Replacing $x$ by $-x$ reflects in the $y$-axis.
$y = 3^x$ and $y = \log_3 x$ are reflections of each other in:
They are inverse functions.
$y = 2^{x+3}$ can be written $y = c \times 2^x$. Find $c$.
$2^{x+3} = 2^3 \times 2^x$.
Find the $y$-intercept of $y = 3 \times 2^x - 5$.
$3 \times 1 - 5 = -2$.
Find the $x$-intercept of $y = \log_3 (x + 2)$.
$x + 2 = 1$.
New numbers every time. Keep going until you get five in a row.