Graphing Trigonometric Functions

Graphing Trigonometric Functions
Section 4.6B Precalculus PreAP/Dual, Revised ©2017 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

§4.6B: Graphing Trig Functions
Review of Graphs 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Transformations Equation: 𝒚=𝑨 𝐭𝐫𝐢𝐠 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝑩 𝒙−𝑪 +𝑫 𝑨 is the amplitude 𝒂: vertically stretches by a factor of 𝒂, 𝟏 𝒂 : Vertically compresses by a factor of 𝟏/𝒂 𝑩 is the period or frequency Period equation: 𝟐𝝅 𝑩 for sine and cosine, 𝝅 𝑩 for tangent 𝑩: phase compresses by a factor of 𝝅 𝑩 𝟏 𝑩 : phasely stretches by a factor of 𝒃 𝑪 is the phase shift If there no GCF taken out, divide the coefficient 𝑬. 𝑫 is the vertical shift F. Frequency is defined as the number of cycles per second 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Steps Identify 𝑨, 𝑩, 𝑪, and 𝑫 from the equation, 𝒚=𝑨 𝐭𝐫𝐢𝐠 𝑩 𝒙−𝑪 +𝑫 Identify the phase shift Period: 𝟐𝜋 𝑩 or 𝜋 𝑩 (for Tan and Cot only) Use the period to identify the spacing 1. Anchor Point Equation: 𝑷𝒆𝒓𝒊𝒐𝒅 𝟒 Start with the phase shift as the middle of the trig table (at the origin) and apply the spacing before and after 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Basic Table Points 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝑪 𝟎 𝟏 −𝟏 𝒚= 𝐜𝐨𝐬 𝒙 𝒙 𝒚 𝑪 𝟏 𝟎 −𝟏 𝒚= 𝐭𝐚𝐧 𝒙 𝒙 𝒚 𝑼𝒏𝒅 −𝟏 𝑪 𝟎 𝟏 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Basic Table Points 𝒚= 𝐜𝐬𝐜 𝒙 𝒙 𝒚 𝑪 𝑼𝒏𝒅 𝟏 −𝟏 𝒚= 𝐬𝐞𝐜 𝒙 𝒙 𝒚 𝑼𝒏𝒅 𝑪 𝟏 −𝟏 𝒚= 𝐜𝐨𝐭 𝒙 𝒙 𝒚 𝑪 𝑼𝒏𝒅 𝟏 𝟎 −𝟏 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 1 Graph 𝒚=𝐬𝐢𝐧 𝒙+𝟏 in one period and identify amplitude, period, vertical shift, phase shift, domain (entire graph), and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚=𝐬𝐢𝐧(𝒙) C 𝟎 𝟏 −𝟏 𝐬𝐢𝐧 (𝒙) +𝟏 𝟏 𝟐 𝟎 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 1 Graph 𝒚=𝐬𝐢𝐧 𝒙+𝟏 in one period and identify amplitude, period, vertical shift, phase shift, domain (entire graph), and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚=𝐬𝐢𝐧 (𝒙)+𝟏 𝟎 𝟏 𝝅/𝟐 𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 2 Graph 𝒚= 𝟏 𝟐 𝐜𝐨𝐬 𝜽−𝟏 in one period and identify amplitude, period, vertical shift, phase shift, domain (entire graph), and range 𝒚 =𝐜𝐨𝐬(𝜽) C 𝟏 𝟎 −𝟏 𝟏 𝟐 𝐜𝐨𝐬𝛉 𝟏/𝟐 𝟎 −𝟏/𝟐 𝟏 𝟐 𝐜𝐨𝐬𝛉−𝟏 −𝟏/𝟐 −𝟏 −𝟑/𝟐 Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 2 Graph 𝒚= 𝟏 𝟐 𝐜𝐨𝐬 𝜽−𝟏 in one period and identify amplitude, period, vertical shift, phase shift, domain (entire graph), and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝟏 𝟐 𝐜𝐨𝐬𝛉−𝟏 𝟎 −𝟏/𝟐 𝝅/𝟐 −𝟏 𝝅 −𝟑/𝟐 𝟑𝝅/𝟐 𝟐𝝅 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Your Turn Graph 𝒚= 𝟏 𝟒 𝐬𝐢𝐧 𝒕 +𝟏 from −𝟐𝝅, 𝟐𝝅 and identify amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚= 𝟏 𝟒 𝐬𝐢𝐧 𝒕 +𝟏 𝒙 𝒚 𝟎 𝟏 𝝅/𝟐 𝟓/𝟒 𝝅 𝟑𝝅/𝟐 𝟑/𝟒 𝟐𝝅 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 3 Given 𝒚=𝟐 𝐭𝐚𝐧 𝒙+𝟏 from −𝝅, 𝝅 and amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚 =𝐭𝐚𝐧(𝒙) 𝑼𝒏𝒅 −𝟏 𝑪 𝟎 𝟏 𝒚=𝟐 𝐭𝐚𝐧 𝒙 𝑼𝒏𝒅 −𝟐 𝟎 𝟐 𝟐 𝐭𝐚𝐧 𝒙+𝟏 𝑼𝒏𝒅 −𝟏 𝟏 𝟑 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 3 Given 𝒚=𝟐 𝐭𝐚𝐧 𝒙+𝟏 from −𝝅, 𝝅 and amplitude, period, vertical shift, and phase shift Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒙 𝟐 𝐭𝐚𝐧 𝒙+𝟏 −𝝅/𝟐 𝑼𝒏𝒅 −𝝅/𝟒 −𝟏 𝟎 𝟏 𝝅/𝟒 𝟑 𝝅/𝟐 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 4 Given 𝒚= 𝟏 𝟐 𝐜𝐬𝐜 𝟐𝒙+𝟏 from −𝟐𝝅, 𝟐𝝅 and amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚 =𝐜𝐬𝐜(𝒙) 𝑪 𝑼𝒏𝒅 𝟏 −𝟏 𝒚= 𝟏 𝟐 𝒄𝒔𝒄 𝒙 𝑼𝒏𝒅 𝟏/𝟐 −𝟏/𝟐 𝟏 𝟐 𝒄𝒔𝒄 𝒙+𝟏 𝑼𝒏𝒅 𝟑/𝟐 𝟏/𝟐 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 4 Given 𝒚= 𝟏 𝟐 𝐜𝐬𝐜 𝟐𝒙+𝟏 from −𝟐𝝅, 𝟐𝝅 and amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒙 𝟏 𝟐 𝒄𝒔𝒄 𝒙+𝟏 𝟎 𝑼𝒏𝒅 𝛑/𝟒 𝟑/𝟐 𝛑/𝟐 𝟑𝛑/𝟒 𝟏/𝟐 𝝅 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Your Turn Given 𝒚= 𝟏 𝟑 𝐬𝐞𝐜 𝒙−𝟏 from −𝟐𝝅, 𝟐𝝅 and identify period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 5 Given 𝒚=𝟓 𝐬𝐢𝐧 𝟐 𝒙− 𝝅 𝟔 +𝟏, identify amplitude, period, vertical shift, phase shift, and points to graph in one period Amplitude Period Vertical Shift 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

𝑩 is OUTSIDE of the parenthesis §4.6B: Graphing Trig Functions
Example 5 Given 𝒚=𝟓 𝐬𝐢𝐧 𝟐 𝒙− 𝝅 𝟔 +𝟏, identify amplitude, period, vertical shift, phase shift, and points to graph in one period 𝑩 is OUTSIDE of the parenthesis Phase Shift 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 5 Given 𝒚=𝟓 𝐬𝐢𝐧 𝟐 𝒙− 𝝅 𝟔 +𝟏, identify amplitude, period, vertical shift, phase shift, and points to graph in one period Phase Shift 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 5 Given 𝒚=𝟓 𝐬𝐢𝐧 𝟐 𝒙− 𝝅 𝟔 +𝟏, identify amplitude, period, vertical shift, phase shift, and points to graph in one period 𝒚= 𝐬𝐢𝐧 𝒙 𝒀=𝟓sin⁡(𝒙) 𝒀=𝟓sin⁡(𝒙)+𝟏 𝒙 𝒚 𝝅/𝟔 𝟎 𝟏 𝟓𝝅/𝟏𝟐 𝟓 𝟔 𝟐𝝅/𝟑 𝟏𝟏𝝅/𝟏𝟐 −𝟏 −𝟓 −𝟒 𝟕𝝅/𝟔 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝑪 𝟎 𝟏 −𝟏 𝒚= 𝐬𝐢𝐧 𝒙 𝒀=𝟓sin⁡(𝒙) 𝒙 𝒚 𝝅/𝟔 𝟎 𝟓𝝅/𝟏𝟐 𝟏 𝟓 𝟐𝝅/𝟑 𝟏𝟏𝝅/𝟏𝟐 −𝟏 −𝟓 𝟕𝝅/𝟔 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝝅/𝟔 𝟎 𝟓𝝅/𝟏𝟐 𝟏 𝟐𝝅/𝟑 𝟏𝟏𝝅/𝟏𝟐 −𝟏 𝟕𝝅/𝟔 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝝅/𝟔 𝟎 𝟓𝝅/𝟏𝟐 𝟏 𝟖𝝅/𝟏𝟐 𝟏𝟏𝝅/𝟏𝟐 −𝟏 𝟏𝟒𝝅/𝟏𝟐 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝝅/𝟔 𝟎 𝟏 −𝟏 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝝅/𝟔 𝟎 𝟓𝝅/𝟏𝟐 𝟏 𝟖𝝅/𝟏𝟐 −𝟏 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝝅/𝟔 𝟎 𝟓𝝅/𝟏𝟐 𝟏 −𝟏 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 6 Given 𝒚=𝟑 𝐜𝐨𝐬 𝟒 𝒙+ 𝝅 𝟐 −𝟏, identify amplitude, period, vertical shift, phase shift, and points to graph in one period Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚= 𝐜𝐨𝐬 𝒙 𝒀=𝟑𝒄𝒐𝒔⁡(𝒙) 𝒀=𝟑𝒄𝒐 𝒔 𝒙 −𝟏 𝒙 𝒚 −𝝅/𝟐 𝟏 𝟑 𝟐 𝟎 −𝟏 𝝅/𝟐 −𝟑 −𝟐 𝝅 𝟑𝝅/𝟐 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 7 Given 𝒚=𝟒 𝐬𝐢𝐧 𝝅𝒙+𝟐 −𝟓, identify amplitude, period, vertical shift, and phase shift and points from −𝟐𝝅, 𝟐𝝅 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 7 Given 𝒚=𝟒 𝐬𝐢𝐧 𝝅𝒙+𝟐 −𝟓, identify amplitude, period, vertical shift, and phase shift and points from −𝟐𝝅, 𝟐𝝅 𝒙 𝑷𝒆𝒓𝒊𝒐𝒅: 𝝅 𝑷𝒉𝒂𝒔𝒆 𝑺𝒉𝒊𝒇𝒕: 𝑳𝒆𝒇𝒕 𝟐/𝝅 𝒚 = sin𝒙 𝒚 = 𝟒 cos (𝝅𝒙 + 𝟐) 𝒚 = 𝟒 cos (𝝅𝒙 + 𝟐) – 𝟓 𝟎 𝟎 – 𝟐/𝝅 𝝅/𝟐 𝝅/𝟒 𝝅/𝟒 – 𝟐/𝝅 𝟏 𝟒 𝝅 𝝅/𝟐 – 𝟐/𝝅 𝟑𝝅/𝟐 𝟑𝝅/𝟒 𝟑𝝅/𝟒 – 𝟐/𝝅 –𝟏 –𝟒 𝟐𝝅 𝝅 – 𝟐/𝝅 𝒙 𝑃𝑒𝑟𝑖𝑜𝑑: 𝜋 𝑷𝒉𝒂𝒔𝒆 𝑺𝒉𝒊𝒇𝒕: 𝑳𝒆𝒇𝒕 𝟐/𝝅 𝒚 = 𝒔𝒊𝒏⁡𝒙 𝒚 = 𝟒 𝒔𝐢𝐧 𝝅𝒙 +𝟐 𝒔𝐢𝐧 𝝅𝒙 +𝟐 𝒚 = 𝟒𝒔𝒊𝒏 𝝅𝒙 +𝟐 𝟒𝒔𝒊𝒏 𝝅𝒙 +𝟐 −𝟓 𝟎 𝟎 – 𝟐/𝝅 –𝟓 𝝅/𝟐 𝝅/𝟒 𝝅/𝟒 – 𝟐/𝝅 𝟏 𝟒 –𝟏 𝝅 𝝅/𝟐 – 𝟐/𝝅 𝟑𝝅/𝟐 𝟑𝝅/𝟒 𝟑𝝅/𝟒 – 𝟐/𝝅 –𝟒 –𝟗 𝟐𝝅 𝝅 – 𝟐/𝝅 x Period: π Phase Shift: Left 2/π y = sin x y = 4 cos (πx + 2) y = 4 cos (πx + 2) – 5 0 – 2/π π/2 π/4 π/4 – 2/π 1 π π/2 – 2/π 3π/2 3π/4 3π/4 – 2/π –1 2π π – 2/π x Period: π Phase Shift: Left 2/π y = sin x y = 4 cos (πx + 2) y = 4 cos (πx + 2) – 5 π/2 π/4 π 3π/2 3π/4 2π x Period: π Phase Shift: Left 2/π y = sin x y = 4 cos (πx + 2) y = 4 cos (πx + 2) – 5 0 – 2/π π/2 π/4 π/4 – 2/π π π/2 – 2/π 3π/2 3π/4 3π/4 – 2/π 2π π – 2/π §4.6B: Graphing Trig Functions 2/22/2019 5:17 PM

Your Turn Given 𝒚=𝟐 𝐬𝐢𝐧 𝟑 𝒙+ 𝝅 𝟑 −𝟓, identify amplitude, period, vertical shift, phase shift, and points to graph in one period Amplitude Period Vertical Shift Phase Shift Spacing (A.P.) Domain Range 𝒚= 𝐬𝐢𝐧 𝒙 𝒀=𝟐𝒔𝒊𝒏⁡(𝒙) 𝒀=𝟐𝒔𝒊𝒏(𝒙)−𝟓 𝒙 𝒚 −𝝅/𝟑 𝟎 −𝟓 −𝝅/𝟔 𝟏 𝟐 −𝟑 𝝅/𝟔 −𝟏 −𝟐 −𝟕 𝝅/𝟑 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 8 Graph 𝒚=−𝟐𝐜𝐨𝐬 𝟑 𝒙+ 𝝅 𝟑 in one period and identify amplitude, period, vertical shift, phase shift, domain, and range 𝒚= 𝐜𝐨𝐬 𝒙 𝒙 𝒚 𝑪 𝟏 𝟎 −𝟏 𝒚= −𝟐𝐜𝐨𝐬 𝟑 𝒙 𝒙 𝒚 −𝝅/𝟑 −𝟐 −𝝅/𝟔 𝟎 𝟐 𝝅/𝟔 𝝅/𝟑 𝒙 −𝝅/𝟑 −𝝅/𝟔 𝟎 𝝅/𝟔 𝝅/𝟑 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 8 Graph 𝒚=−𝟐𝐜𝐨𝐬 𝟑 𝒙+ 𝝅 𝟑 in one period and identify amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Domain Range 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 9 Graph 𝒚=𝟐 𝐬𝐢𝐧 𝟒𝒙+𝝅 −𝟏 from 𝟎,𝟐𝝅 and identify amplitude, period, vertical shift, phase shift, domain, and range 𝒚= 𝟐𝐬𝐢𝐧 𝟒 𝒙+ 𝝅 𝟒 𝒙 𝒚 −𝝅/𝟒 𝟎 −𝝅/𝟖 𝟐 𝝅/𝟖 −𝟐 𝝅/𝟒 𝒚= 𝐬𝐢𝐧 𝒙 𝒙 𝒚 𝑪 𝟎 𝟏 −𝟏 𝒚= 𝟐𝐬𝐢𝐧 𝟒 𝒙+ 𝝅 𝟒 −𝟏 𝒙 𝒚 −𝝅/𝟒 −𝟏 −𝝅/𝟖 𝟏 𝟎 𝝅/𝟖 −𝟑 𝝅/𝟒 𝒙 −𝝅/𝟒 −𝝅/𝟖 𝟎 𝝅/𝟖 𝝅/𝟒 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 9 Graph 𝒚=𝟐 𝐬𝐢𝐧 𝟒𝒙+𝝅 −𝟏 from 𝟎,𝟐𝝅 and identify amplitude, period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Anchor Points Domain Range 𝒚= 𝟐𝐬𝐢𝐧 𝟒 𝒙+ 𝝅 𝟒 −𝟏 𝒙 𝒚 −𝝅/𝟒 −𝟏 −𝝅/𝟖 𝟏 𝟎 𝝅/𝟖 −𝟑 𝝅/𝟒 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 10 Given 𝒚=−𝟐 𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 from −𝝅, 𝝅 and amplitude, period, vertical shift, phase shift, domain, and range 𝒚= −𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 𝒙 𝒚 −𝝅 𝑼𝒏𝒅 −𝟑𝝅/𝟒 𝟏 −𝝅/𝟐 −𝟏 −𝝅/𝟒 −𝟑 𝟎 𝒚= −𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 𝒙 𝒚 −𝝅 𝑼𝒏𝒅 −𝟑𝝅/𝟒 𝟐 −𝝅/𝟐 𝟎 −𝝅/𝟒 −𝟐 𝒚= 𝐭𝐚𝐧 𝒙+𝝅/𝟐 𝒙 𝒚 𝑼𝒏𝒅 −𝟏 𝑪 𝟎 𝟏 𝒙 −𝝅 −𝟑𝝅/𝟒 −𝝅/𝟐 −𝝅/𝟒 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 10 Given 𝒚=−𝟐 𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 from −𝝅, 𝝅 and identify period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Anchor Points Domain Range 𝒚= −𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 𝒙 𝒚 −𝝅 𝑼𝒏𝒅 −𝟑𝝅/𝟒 𝟏 −𝝅/𝟐 −𝟏 −𝝅/𝟒 −𝟑 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Example 10 Given 𝒚=−𝟐 𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 from −𝝅, 𝝅 and amplitude, period, vertical shift, phase shift, domain, and range 𝒚= −𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 −𝟏 𝒙 𝒚 −𝝅 𝑼𝒏𝒅 −𝟑𝝅/𝟒 𝟏 −𝝅/𝟐 −𝟏 −𝝅/𝟒 𝟑 𝟎 𝒚= −𝟐𝐭𝐚𝐧 𝒙+ 𝝅 𝟐 𝒙 𝒚 −𝝅 𝑼𝒏𝒅 −𝟑𝝅/𝟒 𝟐 −𝝅/𝟐 𝟎 −𝝅/𝟒 −𝟐 𝒚= 𝐭𝐚𝐧 𝒙+𝝅/𝟐 𝒙 𝒚 𝑼𝒏𝒅 −𝟏 𝑪 𝟎 𝟏 𝒙 −𝝅 −𝟑𝝅/𝟒 −𝝅/𝟐 −𝝅/𝟒 𝟎 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Your Turn Given 𝒚= 𝐬𝐞𝐜 𝟐 𝒙+𝝅 +𝟐 from −𝟐𝝅, 𝟐𝝅 and identify period, vertical shift, phase shift, domain, and range Amplitude Period Vertical Shift Phase Shift Anchor Points Domain Range 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Assignment Worksheet 2/22/2019 5:17 PM §4.6B: Graphing Trig Functions

Graphing Trigonometric Functions

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Graphing Trigonometric Functions

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