How to Graph the Cosine Function?
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 \(y=A\cos(Bx)+D\), amplitude is \(|A|\), period is \(2\pi/|B|\), 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
\(y=A\cos(Bx)+D\)
\(\text{amplitude}=|A|\)
\(\text{period}=\frac{2\pi}{|B|}\)
\(\text{midline}=y=D\)
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: \(y=2\cos(4x)+3\)
- 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: \((0,1),(\pi/2,0),(\pi,-1),(3\pi/2,0),(2\pi,1)\).
Try one before moving on
Try: Find the amplitude and period of \(y=3\cos(6x)\).
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 \(y=5\cos x\), 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 \(\frac {3\pi}{2}\),\(y\) increases from \(-1\) to \(0\).
- By increasing \(x\) from \(\frac {3\pi}{2}\) to \(2\pi\), \(y\) 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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