Sine and Cosine Graphs Amplitude, midline, and period 

𝑓(𝑥) = 𝑎 𝑠𝑖𝑛(𝐵(𝑥 – ℎ)) + 𝑘 𝑓(𝑥) = 𝑎 𝑐𝑜𝑠(𝐵(𝑥 – ℎ)) + 𝑘 • Amplitude: |𝑎| The height of the curve above and the depth below the midline • Midline: 𝑦 = 𝑘 • Period: 34 |5| 𝐵 = 2𝜋 |𝑃| • Horizontal Shift: ℎ • Quarter Split: 9:;<=> ? Average y value of the curve Length of the interval in which one complete curve appears One quarter of a full cycle (period) of a graph (helps in graphing) 

Graphing notes 𝑓(𝑥) = 𝑎𝑚𝑝 𝑠𝑖𝑛( 2𝜋 𝑝 𝑥) + 𝑚𝑖𝑑𝑙𝑖𝑛𝑒 Sine • Starts at midline • If positive it goes up first • If negative it goes down first 𝑓(𝑥) = 𝑎𝑚𝑝 𝑐𝑜𝑠( 2𝜋 𝑝 𝑥) + 𝑚𝑖𝑑𝑙𝑖𝑛𝑒 Cosine • If positive it starts high • If negative it starts low 

5 𝑦 = −2 𝑃 = 2𝜋 2 = 𝜋 𝜋 𝑄𝑆 = 𝜋 4 𝜋 4 5 reference points 𝜋 4 𝜋 2 3𝜋 4 𝜋 -2 -8 3 Sine: start at midline Negative: go down first 7 𝑦 = 4 𝑃 = 2𝜋 1 = 2𝜋 2𝜋 𝑄𝑆 = 2𝜋 4 = 𝜋 2 𝜋 2 𝜋 2 3𝜋 2 𝜋 2𝜋 4 11 -3 Negative cosine: start low period 

State the amplitude, midline, period, and quarter split of the following functions 3 𝑦 = −7 𝑃 = 2𝜋 1 = 2𝜋 2𝜋 𝑄𝑆 = 2𝜋 4 = 𝜋 2 𝜋 2 2 𝑦 = 0 𝑃 = 2𝜋 2 = 𝜋 𝑄𝑆 = 𝜋 4 𝜋 𝜋 4 2 𝑦 = 4 𝑃 = 2𝜋 5 2𝜋 5 𝑄𝑆 = 2𝜋 5 ∗ 1 4 = 𝜋 10 𝜋 10 Instead of dividing by a fraction, multiply by the reciprocal 9 𝑦 = 2 𝑃 = 2𝜋 3 2𝜋 3 𝑄𝑆 = 2𝜋 3 ∗ 1 4 = 𝜋 6 𝜋 6 

1 𝑦 = 2 𝑃 = 2𝜋 4 = 𝜋 2 𝜋 2 𝑄𝑆 = 𝜋 2 ∗ 1 4 = 𝜋 8 𝜋 8 1 𝑦 = −5 𝑃 = 2𝜋 2 = 𝜋 𝜋 𝑄𝑆 = 𝜋 4 𝜋 4 3 𝑦 = 0 𝑃 = 2𝜋 1 2 4𝜋 𝑄𝑆 = 4𝜋 4 = 𝜋 𝜋 Instead of dividing by a fraction, multiply by the reciprocal 4 𝑦 = 0 𝑃 = 2𝜋 1 3 = 2𝜋 ∗ 3 1 = 6𝜋 6𝜋 𝑄𝑆 = 6𝜋 4 = 3𝜋 2 3𝜋 2 = 2𝜋 ∗ 2 1 = 4𝜋 

State the amplitude, midline, period, and quarter split of the following functions 𝑦 = 0 3 Period starts and ends in the same position, passing through the 5 reference points 2 𝑓(𝑥) = sin( x) 3 − 𝐵 = 2𝜋 2 = 𝜋 𝜋 𝐵 = 2𝜋 |𝑃| 𝑓(𝑥) = 𝑎𝑚𝑝 𝑠𝑖𝑛( 2𝜋 𝑝 𝑥) + 𝑚𝑖𝑑𝑙𝑖𝑛𝑒 .5 -.5 .5 𝑦 = 0 𝐵 = 2𝜋 4 = 𝜋 2 4 𝑓(𝑥) = 𝑎𝑚𝑝 𝑐𝑜𝑠( 2𝜋 𝑝 𝑥) + 𝑚𝑖𝑑𝑙𝑖𝑛𝑒 𝑓 𝑥 = .5𝑐𝑜𝑠( 𝜋 2 𝑥) 

2 -4 -1 𝑦 = 3 1 2𝜋 𝜋 3 𝑦 = −1 𝑓 𝑥 = 𝑠𝑖𝑛 𝑥 + 3 𝑓 𝑥 = −3𝑐𝑜𝑠 2𝑥 − 1 𝐵 = 2𝜋 2𝜋 = 1 𝐵 = 2𝜋 𝜋 = 2