How to Graph the Cosine Function?
How to graph the cosine function, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
A step-by-step guide to graph the cosine function
From the graph, we can know how (x) and (y) change:
Tutor-style math help
Graph the Cosine Function: what to notice and how to work it
Trigonometry skill
A cosine graph is a repeating wave that starts at a high point on the parent graph. Mark amplitude, period, midline, and five key points before sketching.
What to notice first
For (y=Acos(Bx)+D), amplitude is (|A|), period is (2pi/|B|), and midline is (y=D).
Common student mistake
Do not use sine's starting point automatically. Parent cosine starts at ((0,1)), not ((0,0)).
Key formulas and cues
(y=Acos(Bx)+D)
(text{amplitude}=|A|)
(text{period}=frac{2pi}{|B|})
(text{midline}=y=D)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Read a cosine rule
Example: (y=2cos(4x)+3)
- Amplitude is |2|.
- Period is 2pi/4 = pi/2.
- Midline is y = 3.
Answer: Amplitude (2), period (pi/2), midline (y=3).
Place parent cosine points
Example: Graph one cycle of (y=cos x).
- Start at (0, 1).
- Use quarter-period steps.
- The y-values are 1, 0, -1, 0, 1.
Answer: ((0,1),(pi/2,0),(pi,-1),(3pi/2,0),(2pi,1)).
Try one before moving on
Try: Find the amplitude and period of (y=3cos(6x)).
Answer: Amplitude (3), period (pi/3).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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Graph the Cosine Function: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. For (y=5cos x), the amplitude is:
2. For (y=cos(2x)), the period is:
3. The parent cosine graph starts at:
- By increasing (x) from (0) to (frac{pi}{2}), (y) decreases from (1) to (0).
- By increasing (x) from (frac{pi}{2}) to (pi), (y) decreases from (0) to (-1).
- By increasing (x) from (pi) to (frac {3pi}{2}),(y) increases from (-1) to (0).
- By increasing (x) from (frac {3pi}{2}) to (2pi), (y) increases from (0) to (1).
This pattern repeats itself when we plot a larger subset of the domain of the (cos) function. For example, add to the points given above the point whose (x)-coordinates are in the interval (−2π≤x≤0):
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