Curve sketching
This PowerPoint presentation shows
the different stages involved in
sketching the graph
4  2x
y
2 x
4  2x
Sketching the graph y 
2 x
Step 1: Find where the graph cuts the axes
When x = 0, y = 2, so the
graph goes through the
point (0, 2).
When y = 0, x = 2, so the
graph goes through the
point (2, 0).
4  2x
Sketching the graph y 
2 x
Step 2: Find the vertical asymptotes
The denominator is zero
when x = -2
The vertical
asymptote is x = -2
4  2x
Sketching the graph y 
2 x
Step 2: Find the vertical asymptotes
The denominator is zero
when x = -2
The vertical
asymptote is x = -2
For now, don’t worry about the behaviour of the graph near
the asymptotes. You may not need this information.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
For large positive values of x, y is slightly greater than -2.
So as x → ∞, y → -2 from above.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
For large positive values of x, y is slightly greater than -2.
So as x → ∞, y → -2 from above.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
For large negative values of x, y is slightly less than -2.
So as x → -∞, y → -2 from below.
4  2x
Sketching the graph y 
2 x
Step 3: Examine the behaviour as x tends to infinity
For numerically large
values of x, y → -2.
This means that y = -2
is a horizontal
asymptote.
For large negative values of x, y is slightly less than -2.
So as x → -∞, y → -2 from below.
4  2x
Sketching the graph y 
2 x
Step 4: Complete the sketch
Since the graph only
crosses the x axis at (2, 0),
we can complete the part
of the graph to the left of
the asymptote.
4  2x
Sketching the graph y 
2 x
Step 4: Complete the sketch
Since the graph only
crosses the x axis at (2, 0),
we can complete the part
of the graph to the left of
the asymptote.
4  2x
Sketching the graph y 
2 x
Step 4: Complete the sketch
We can also complete the
part of the graph to the
right of the asymptote,
using the points where the
graph cuts the axes.
4  2x
Sketching the graph y 
2 x
Step 4: Complete the sketch
We can also complete the
part of the graph to the
right of the asymptote,
using the points where the
graph cuts the axes.
Notice that in fact we did not need to know whether the
graph was above or below the horizontal asymptote for
numerically large x. The sketch shows the only possibility!