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This unit covers sine and cosine graphs, tangent graphs and amplitude period and phase shift — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers sine and cosine graphs, tangent graphs and amplitude period and phase shift — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Sine And Cosine Graphs
The sine function starts at (0, 0) and the cosine function starts at (0, 1), both oscillating between -1 and 1 with a period of 2π. Students must be able to identify key points (maximum, minimum, zeros) and sketch one full cycle from a given equation. Transformations shift, stretch, or reflect the basic shape.
Key Points
- sin(x) passes through (0,0), peaks at (π/2, 1), returns to 0 at (π), troughs at (3π/2, -1), completes at (2π, 0)
- cos(x) passes through (0,1), hits zero at (π/2), troughs at (π, -1), hits zero at (3π/2), completes at (2π, 1)
- Both have domain: all real numbers; range: [-1, 1]; period: 2π
- Reflecting over x-axis (negative leading coefficient) flips all y-values
Sketch one full period of y = -cos(x) and identify all key points.
Start with the standard cosine key points: (0,1), (π/2,0), (π,-1), (3π/2,0), (2π,1). Because of the negative sign, flip all y-values: (0,-1), (π/2,0), (π,1), (3π/2,0), (2π,-1). The graph is a cosine curve reflected over the x-axis, starting at a minimum instead of a maximum.
2 Tangent Graphs
The tangent function has a period of π (not 2π), vertical asymptotes where cosine equals zero, and passes through the origin within each cycle. Students must locate asymptotes, plot the three key points per cycle, and recognize the basic S-shaped curve between asymptotes.
Key Points
- Vertical asymptotes occur at x = π/2 + nπ for any integer n
- Key points within one cycle (−π/2 to π/2): (−π/4, −1), (0, 0), (π/4, 1)
- Period is π; the graph repeats every π units
- tan(x) has no amplitude — the range is all real numbers
State the equations of two consecutive vertical asymptotes of y = tan(x) and identify the x-intercept between them.
The asymptotes are at x = -π/2 and x = π/2, which are the two closest asymptotes on either side of the origin. The x-intercept falls exactly halfway between them at x = 0, giving the point (0, 0). This midpoint-is-zero pattern holds for every cycle of the tangent graph.
3 Amplitude
Amplitude is the vertical stretch factor of a sine or cosine function, equal to |A| in y = A·sin(x) or y = A·cos(x). It defines the maximum distance from the midline to a peak or trough. Tangent has no amplitude.
Key Points
- Amplitude = |A|; the range of y = A·sin(x) becomes [−|A|, |A|]
- A negative value of A reflects the graph over the x-axis but does NOT change the amplitude
- The midline stays at y = 0 unless a vertical shift D is added
- On a graph, amplitude = (max value − min value) ÷ 2
What is the amplitude of y = -3sin(x), and what is its range?
The coefficient A = -3, so the amplitude is |-3| = 3. The negative sign flips the graph but does not affect amplitude. The range is [-3, 3], meaning the graph reaches a low of -3 and a high of 3.
4 Period And Phase Shift
In the form y = A·sin(Bx − C) + D, the period equals 2π/|B| for sine and cosine (or π/|B| for tangent), and the phase shift equals C/B. Students must calculate both values from the equation and use them to correctly position the graph on the x-axis.
Key Points
- Period of sin/cos: 2π/|B|; Period of tan: π/|B|
- Phase shift = C/B; positive result shifts right, negative shifts left
- Always rewrite in the form y = A·sin(B(x − h)) + D to read phase shift as h directly
- Vertical shift D moves the midline from y = 0 to y = D
Find the period and phase shift of y = 2sin(3x − π).
Here B = 3 and C = π, so the period is 2π/3. The phase shift is C/B = π/3, and since it is positive the graph shifts π/3 units to the right. To confirm, factor the argument: y = 2sin(3(x − π/3)), which clearly shows a rightward shift of π/3.
Questions, answered.
What is Graphing Trig Functions?
Graphing Trig Functions is Unit 4 of Trigonometry, covering sine and cosine graphs, tangent graphs and amplitude period and phase shift.
How to study for Trigonometry Unit 4?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.