What is the asymptote of y = eˣ + 4?

y = 4. The whole curve is moved up 4.

What is the asymptote of y = ln(x − 5)?

x = 5. The log needs x − 5 to be positive.

How are the graphs of y = eˣ and y = ln x related?

Each is the reflection of the other in the line y = x. They are inverse functions.

Graphs of exponential and logarithmic functions · Cambridge IGCSE Additional Mathematics

In plain words

y = eˣ climbs faster and faster and never touches the x-axis. y = ln x is the same curve reflected in the line y = x: it creeps upwards and never touches the y-axis. Every graph in this topic is one of those two, stretched or moved.

6 things to know

y = eˣ passes through (0, 1). It is always positive, and the x-axis (y = 0) is an asymptote.
y = ln x passes through (1, 0). It exists only for x > 0, and the y-axis (x = 0) is an asymptote.
eˣ and ln x are inverses of each other, so each graph is the reflection of the other in y = x.
y = ke^(nx) + a has the horizontal asymptote y = a. It crosses the y-axis at k + a.
y = k ln(ax + b) has a vertical asymptote where ax + b = 0. It crosses the x-axis where ax + b = 1.
An asymptote is a line the curve gets closer and closer to but never reaches. Give its equation.

Worked example

For y = 2eˣ − 6, find the asymptote and the points where the curve meets the axes.

As x becomes large and negative, eˣ approaches 0, so y approaches −6. The asymptote is y = −6.
At x = 0: y = 2 − 6 = −4. The curve crosses the y-axis at (0, −4).
At y = 0: 2eˣ = 6, so eˣ = 3 and x = ln 3. It crosses the x-axis at (ln 3, 0).

Worked example

For y = ln(2x − 4), find the asymptote and where the curve crosses the x-axis.

The log needs 2x − 4 > 0, so x > 2. The asymptote is x = 2.
y = 0 when 2x − 4 = 1, so x = 2.5.

Tips and tricks

State an asymptote as an equation, such as y = −6 or x = 2, not just "the x-axis moved down".
ln of 1 is 0, so a log graph crosses the x-axis where the bracket equals 1, not where it equals 0.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Graphs of exponential and logarithmic functions: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

Through which point does y = eˣ pass?
- (0, 0)
- (0, 1) (the answer)
- (1, 0)
- (1, 1)
e⁰ = 1.
Through which point does y = ln x pass?
- (0, 0)
- (0, 1)
- (1, 0) (the answer)
- (e, 0)
ln 1 = 0.
What is the asymptote of y = eˣ + 4?
- y = 0
- y = 4 (the answer)
- x = 4
- x = 0
The whole curve is moved up 4.
What is the asymptote of y = ln(x − 5)?
- y = 5
- x = 0
- x = 5 (the answer)
- y = 0
The log needs x − 5 to be positive.
How are the graphs of y = eˣ and y = ln x related?
- They are the same curve.
- Each is the reflection of the other in the line y = x. (the answer)
- Each is the reflection of the other in the x-axis.
- They never both exist.
They are inverse functions.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

2 questions, 6 marks. Write your answers on paper, then check them.