Worked examples
The question
Question: graph y = sec x.
y = sec x
1 Write it as 1 over
sec x means 1 over cos x.
sec goes with cos: each one is cos turned upside down.
sec x = 1cos x
Careful!
sec x is 1 over cos x.
It is not the inverse cosine, cos to the minus 1.
That is a different thing.
Careful: sec x = 1 ÷ cos x, not cos−1 x
2 Draw the helper graph
Draw the grid.
A mark every π/2: a quarter of the period.
Our helper is y = cos x.
Its amplitude is 1, and its period is 2π over 1: 2π.
y = cos x
period = 2π1 = 2π
We draw the helper wave lightly, dashed.
sec will be built from it.
3 Find the gaps
Where the helper is 0, sec is 1/0.
We can never divide by 0: there is a gap.
We draw a dashed line there: an asymptote.
The helper crosses the x-axis at π/2 and 3π/2, and so on.
cos x = 0: x = π2, 3π2, ...
4 Find the turning points
Where the helper is at its top or bottom, cos x is 1 or −1.
1/1 is 1, and 1 over −1 is −1: the value stays the same.
sec touches the helper there.
1 ÷ 1 = 1, 1 ÷ (−1) = −1
5 Draw the U shapes
Between two gaps, the graph is a U.
It starts at the turning point.
As the helper gets closer to 0, 1 over it gets bigger and bigger: up to the gap.
Where the helper is below the axis, the U opens down.
6 Check
Let's check one more point: x = π/3.
cos π/3 is one half.
1 over one half is 2.
So y is 2: the point is on the U.
Right!
x = π3: cos π3 = 12, y = 1 ÷ 12 = 2
Let's look back.
First: write it as 1 over.
Then: draw the helper graph.
Then: find the gaps.
Then: find the turning points.
Then: draw the U shapes.
Last: check.
▶ Watch it solved
The question
Question: graph y = 2 sec x.
y = 2 sec x
1 Write it as 1 over
sec x means 1 over cos x.
sec goes with cos: each one is cos turned upside down.
sec x = 1cos x
So y = 2 over cos x.
y = 2cos x
Careful!
sec x is 1 over cos x.
It is not the inverse cosine, cos to the minus 1.
That is a different thing.
Careful: sec x = 1 ÷ cos x, not cos−1 x
2 Draw the helper graph
Draw the grid.
A mark every π/2: a quarter of the period.
Our helper is y = 2 cos x.
Its amplitude is 2, and its period is 2π over 1: 2π.
y = 2 cos x
period = 2π1 = 2π
We draw the helper wave lightly, dashed.
sec will be built from it.
3 Find the gaps
Where the helper is 0, sec is 1/0.
We can never divide by 0: there is a gap.
We draw a dashed line there: an asymptote.
The helper crosses the x-axis at π/2 and 3π/2, and so on.
cos x = 0: x = π2, 3π2, ...
4 Find the turning points
Where the helper is at its top or bottom, cos x is 1 or −1.
1/1 is 1, and 1 over −1 is −1: the value stays the same.
sec touches the helper there.
1 ÷ 1 = 1, 1 ÷ (−1) = −1
5 Draw the U shapes
Between two gaps, the graph is a U.
It starts at the turning point.
As the helper gets closer to 0, 1 over it gets bigger and bigger: up to the gap.
Where the helper is below the axis, the U opens down.
6 Check
Let's check one more point: x = π/3.
cos π/3 is one half.
1 over one half is 2, times 2 is 4.
So y is 4: the point is on the U.
Right!
x = π3: cos π3 = 12, y = 2 × 1 ÷ 12 = 4
Let's look back.
First: write it as 1 over.
Then: draw the helper graph.
Then: find the gaps.
Then: find the turning points.
Then: draw the U shapes.
Last: check.
▶ Watch it solved
The question
Question: graph y = −sec x.
y = −sec x
1 Write it as 1 over
sec x means 1 over cos x.
sec goes with cos: each one is cos turned upside down.
sec x = 1cos x
So y = -1 over cos x.
y = −1cos x
Careful!
sec x is 1 over cos x.
It is not the inverse cosine, cos to the minus 1.
That is a different thing.
Careful: sec x = 1 ÷ cos x, not cos−1 x
2 Draw the helper graph
Draw the grid.
A mark every π/2: a quarter of the period.
Our helper is y = −cos x.
Its amplitude is 1, and its period is 2π over 1: 2π.
y = −cos x
period = 2π1 = 2π
We draw the helper wave lightly, dashed.
sec will be built from it.
3 Find the gaps
Where the helper is 0, sec is 1/0.
We can never divide by 0: there is a gap.
We draw a dashed line there: an asymptote.
The helper crosses the x-axis at π/2 and 3π/2, and so on.
cos x = 0: x = π2, 3π2, ...
4 Find the turning points
Where the helper is at its top or bottom, cos x is 1 or −1.
1/1 is 1, and 1 over −1 is −1: the value stays the same.
sec touches the helper there.
1 ÷ 1 = 1, 1 ÷ (−1) = −1
5 Draw the U shapes
Between two gaps, the graph is a U.
It starts at the turning point.
As the helper gets closer to 0, 1 over it gets bigger and bigger: up to the gap.
Where the helper is below the axis, the U opens down.
6 Check
Let's check one more point: x = π/3.
cos π/3 is one half.
1 over one half is 2, times -1 is -2.
So y is -2: the point is on the upside-down U.
Right!
x = π3: cos π3 = 12, y = -1 × 1 ÷ 12 = -2
Let's look back.
First: write it as 1 over.
Then: draw the helper graph.
Then: find the gaps.
Then: find the turning points.
Then: draw the U shapes.
Last: check.
▶ Watch it solved
