Worksheet for practicing sketching parent graphs of trigonometric functions and identifying their key features.

Let’s go step by step to solve each part of the worksheet. We’ll focus on matching trig functions to their key features.

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First, recall parent graph properties:

- Sine (sin):
- Range: [-1, 1]
- Period: 2π
- Zeros at ±π(k) → k = integer
- y-intercept: 0
- Continuous wave shape
- Relative max at π/2 + 2πk → value 1
- Relative min at 3π/2 + 2πk → value -1

- Cosine (cos):
- Range: [-1, 1]
- Period: 2π
- Zeros at ±π/2 ± π(k) → actually zeros at π/2 + πk
- y-intercept: 1
- Absolute max at 0 + 2πk → value 1
- Absolute min at π + 2πk → value -1
- Continuous wave shape

- Tangent (tan):
- Range: (-∞, ∞)
- Period: π
- Asymptotes at ±π/2 ± π(k) → odd multiples of π/2
- No absolute max/min
- Zeros at ±π(k)
- Domain: all reals except asymptotes → not (-∞, ∞)
- Alternating u-shapes between asymptotes

- Cosecant (csc = 1/sin):
- Range: (-∞, -1] ∪ [1, ∞)
- Period: 2π
- Asymptotes where sin=0 → at ±π(k)
- No y-intercept (undefined at x=0)
- Relative min of 1 (at peaks of sin), relative max of -1 (at troughs of sin)
- Alternating u-shapes

- Secant (sec = 1/cos):
- Range: (-∞, -1] [1, ∞)
- Period: 2π
- Asymptotes where cos=0 → at ±π/2 ± π(k)
- y-intercept: 1 (since cos(0)=1 → sec(0)=1)
- Absolute min of 1, absolute max of -1? Wait — no: since it's reciprocal, when cos=1, sec=1 (min positive); when cos=-1, sec=-1 (max negative). But “absolute” usually means global extreme values — so yes, absolute min is 1, absolute max is -1? Actually, in terms of magnitude, but conventionally we say:
- Absolute minimum value: 1 (positive side)
- Absolute maximum value: -1 (negative side) — though this is confusing because -1 < 1. Better to think:
The function never takes values between -1 and 1. So its *minimum* output is -∞, but that’s not helpful. In context of "absolute max/min" for these problems, they mean the closest points to zero:
→ For csc and sec:
- Absolute min of 1 (meaning smallest positive value)
- Absolute max of -1 (largest negative value)
This matches typical textbook usage.

- Cotangent (cot = 1/tan):
- Range: (-∞, ∞)
- Period: π
- Asymptotes at ±π(k) → where tan=0
- Zeros at ±π/2 ± π(k) → where tan undefined → cot=0 there
- No y-intercept? At x=0, tan(0)=0 → cot(0) undefined → so no y-intercept
- Alternating u-shapes
- Continuous? No — has asymptotes

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Now let’s match each feature:

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7) No y-intercept:
Which functions are undefined at x=0?
→ tan(0)=0 → defined → has y-intercept 0
→ cot(0) undefined → no y-intercept
→ csc(0) undefined → no y-intercept
→ sec(0)=1 → has y-intercept
→ sin(0)=0 → has y-intercept
→ cos(0)=1 → has y-intercept
✔ So: cot, csc

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8) y-intercept of 1:
At x=0, which equal 1?
→ cos(0)=1 → yes
→ sec(0)=1 → yes
Others: sin(0)=0, tan(0)=0, csc(0) undef, cot(0) undef
✔ So: cos, sec

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9) Absolute max of 1:
This means the highest point the graph reaches is 1.
→ sin: goes up to 1 → yes
→ cos: goes up to 1 → yes
→ tan, cot: go to → no
→ csc, sec: go to ±∞, but also touch 1 and -1 — but “absolute max” would be ∞? Not really. In context, for bounded functions only.
Actually, for csc/sec, they don’t have an absolute max because they go to infinity. Only sin and cos are bounded above by 1.
But wait — question says “Absolute max of 1” — meaning the maximum value attained is exactly 1.
→ sin and cos both attain 1 and never exceed it → ✔
→ csc and sec attain 1 but also go to ∞ → so their absolute max is ∞, not 1 → ✘
So only sin and cos? But let’s check #10 too.

Wait — perhaps for csc/sec, “absolute max of -1” makes sense as the least negative? Let’s hold.

Actually, standard interpretation in such worksheets:
- For sin/cos: absolute max = 1, absolute min = -1
- For csc/sec: they have local mins/maxes at 1 and -1, but globally unbounded → so “no abs max/min” or sometimes they refer to the vertex values.

Looking ahead to #13: “No abs. max or min” — that should be tan, cot, csc, sec? But csc/sec do have relative extrema.

Better to use common textbook answers:

Typically:
- sin, cos: have absolute max 1, absolute min -1
- tan, cot: no absolute max or min (go to ±∞)
- csc, sec: no absolute max or min (also go to ±∞), BUT they have relative max/min at ±1

But look at #9 and #10:

#9: Absolute max of 1 → must be functions that reach 1 and never go higher → sin, cos
#10: Absolute min of -1 → same → sin, cos

Then #11: Relative max of -1 → this would be for csc/sec? Because at their lowest humps, they hit -1 (for example, csc has a downward U reaching -1)

Similarly, #12: Relative min of 1 → upward U reaching 1

And #13: No abs. max or min → tan, cot, csc, sec — since all go to infinity

But let’s verify with known patterns.

I recall that in many curricula:

- sin, cos: bounded → abs max 1, abs min -1
- tan, cot: unbounded → no abs max/min
- csc, sec: unbounded → no abs max/min, but have relative extrema at ±1

So:

#9: Absolute max of 1 → sin, cos
#10: Absolute min of -1 → sin, cos
#11: Relative max of -1 → csc, sec (because their lower branches peak at -1)
#12: Relative min of 1 → csc, sec (upper branches bottom out at 1)
#13: No abs. max or min → tan, cot, csc, sec (all unbounded)

Yes, that fits.

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14) Asymptotes at ±π(k):
Asymptotes at integer multiples of π → where?
→ csc: undefined where sin=0 → at 0, ±π, ±2π... → yes
→ cot: undefined where tan=0 → same places → yes
→ tan: asymptotes at ±π/2 ± πk → not at πk
→ sec: asymptotes at ±π/2 ± πk
→ sin, cos: no asymptotes
✔ So: csc, cot

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15) Asymptotes at ±π/2 ± π(k):
That’s odd multiples of π/2 →
→ tan: yes
→ sec: yes (where cos=0)
→ others: no
✔ So: tan, sec

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16) Domain (-∞, ∞):
Defined everywhere?
→ sin, cos: yes
→ tan: no (asymptotes)
→ cot: no
→ csc: no
→ sec: no
✔ So: sin, cos

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17) Range (-∞, ∞):
All real numbers?
→ tan: yes
→ cot: yes
→ others: bounded or restricted
✔ So: tan, cot

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18) Range (-∞, -1] ∪ [1, ∞):
Never between -1 and 1?
→ csc: yes (since |sin| ≤ 1 → |csc| ≥ 1)
→ sec: same
✔ So: csc, sec

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19) Range [-1, 1]:
Bounded between -1 and 1 inclusive?
→ sin: yes
→ cos: yes
✔ So: sin, cos

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20) Period of π:
Repeats every π?
→ tan: yes
→ cot: yes
→ others: 2π
✔ So: tan, cot

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21) Period of 2π:
→ sin, cos, csc, sec
✔ So: sin, cos, csc, sec

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22) No zeros:
Where does the function never cross x-axis?
→ csc: 1/sin → never zero (numerator always 1)
→ sec: 1/cos → never zero
→ sin: zeros at πk
→ cos: zeros at π/2 + πk
→ tan: zeros at πk
→ cot: zeros at π/2 + πk
✔ So: csc, sec

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23) Zeros at ±π/2 ± π(k):
That’s ..., -3π/2, -π/2, π/2, 3π/2, ...
→ cos: zeros there? cos(π/2)=0, cos(3π/2)=0 → yes
→ cot: cot(x) = cos(x)/sin(x) → zero when cos(x)=0 → same places → yes
Check:
At x=π/2: cos(π/2)=0 → cot(π/2)=0/sin(π/2)=0/1=0 → yes
So both cos and cot have zeros at those points?
But cos is continuous, cot has asymptotes elsewhere.

Wait — the pattern ±π/2 ± π(k) is the same as π/2 + πk for integer k.

Zeros of cos: at π/2 + πk → yes
Zeros of cot: also at π/2 + πk → yes
Are there others?
sin: zeros at πk → no
tan: zeros at πk → no
csc, sec: no zeros → no
✔ So: cos, cot

But let me double-check cot:
cot(x) = 0 when cos(x)=0 and sin(x)≠0 → which is exactly at π/2 + πk → yes.

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24) Zeros at ±π(k):
Integer multiples of π →
→ sin: sin(0)=0, sin(π)=0 → yes
→ tan: tan(0)=0, tan(π)=0 → yes
→ others: no
✔ So: sin, tan

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25) Continuous wave shape:
Smooth, no breaks, periodic waves →
→ sin, cos: yes
→ tan, cot: have asymptotes → not continuous
→ csc, sec: have asymptotes → not continuous
✔ So: sin, cos

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26) Alternating u-shapes:
Between asymptotes, they form U or inverted U shapes alternating →
→ tan: yes (increasing from -∞ to ∞, but shaped like S? Wait — actually tan has increasing curve, not U-shape. Hmm.)

Wait — let’s think:

- tan: between -π/2 and π/2, it goes from -∞ to ∞, passing through 0 — it’s strictly increasing, concave down then up? Not really “u-shape”.

Actually, “alternating u-shapes” typically refers to csc, sec, cot.

Standard description:

- csc: between 0 and π, it has a U-shape opening up from 1 to ∞ then back? No — actually, from 0+ to π-, sin is positive, so csc is positive, starts at +∞, down to 1 at π/2, back to +∞ — so it’s a U-shape opening upwards? But only one per period? And next period (π to 2π) sin negative, so csc negative: from -∞ up to -1 at 3π/2, back to -∞ — so inverted U.

So alternating: up-U, then down-U, etc.

Similarly for sec: same thing.

For cot: between 0 and π, cot goes from +∞ down to -∞, crossing zero at π/2 — so it’s decreasing, not U-shaped.

Actually, cot is monotonic decreasing in each interval — no U-shape.

Tan is monotonic increasing.

So only csc and sec have the “U” and “inverted U” alternating.

Confirming with sources: Yes, csc and sec graphs consist of alternating U-shaped curves (opening up and down).

Cot and tan are straight-ish curves without turning points.

So:

✔ csc, sec

But let’s see the exact wording: “Alternating u-shapes” — implies the shape changes direction, forming cups.

Yes, csc and sec.

Some might include cot, but technically cot doesn't have local min/max in the same way — it’s strictly decreasing.

In each interval, cot decreases from +∞ to -∞ — no vertex.

Whereas csc has a minimum at 1 (in first hump), then maximum at -1 (next hump), etc.

So I think only csc and sec.

Final decision: csc, sec

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Now compile all answers:

7) No y-intercept: cot, csc
8) y-intercept of 1: cos, sec
9) Absolute max of 1: sin, cos
10) Absolute min of -1: sin, cos
11) Relative max of -1: csc, sec
12) Relative min of 1: csc, sec
13) No abs. max or min: tan, cot, csc, sec
14) Asymptotes at ±π(k): csc, cot
15) Asymptotes at ±π/2 ± π(k): tan, sec
16) Domain (-∞, ∞): sin, cos
17) Range (-∞, ∞): tan, cot
18) Range (-∞, -1] ∪ [1, ∞): csc, sec
19) Range [-1, 1]: sin, cos
20) Period of π: tan, cot
21) Period of 2π: sin, cos, csc, sec
22) No zeros: csc, sec
23) Zeros at ±π/2 ± π(k): cos, cot
24) Zeros at ±π(k): sin, tan
25) Continuous wave shape: sin, cos
26) Alternating u-shapes: csc, sec

Note: For #23, some might argue only cot, but cos also has zeros there. The question doesn’t specify “only” or “exclusively”, so if a function has zeros at those points, it qualifies. Cos does, cot does.

Similarly, for #24, sin and tan both have zeros at πk.

All seem consistent.

Final Answer:

7) cot, csc
8) cos, sec
9) sin, cos
10) sin, cos
11) csc, sec
12) csc, sec
13) tan, cot, csc, sec
14) csc, cot
15) tan, sec
16) sin, cos
17) tan, cot
18) csc, sec
19) sin, cos
20) tan, cot
21) sin, cos, csc, sec
22) csc, sec
23) cos, cot
24) sin, tan
25) sin, cos
26) csc, sec