Nat 5 Creation of BASIC Trig Graphs Graphs of the form y = a sin xo

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Graphing Trig Functions

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Trigonometry Graphs www.mathsrevision.com Nat 5 Creation of BASIC Trig Graphs Graphs of the form y = a sin xo Graphs of the form y = a sin bxo Graphs of the form y = a sin bxo + c www.mathsrevision.com Phase angle y = a sin(x + b) Exam Type Questions created by Mr. Lafferty

Trig Graphs Creation of a sine graph Creation of a cosine graph Nat 5 Sine Graph Creation of a sine graph Cosine Graph Creation of a cosine graph Tan Graph www.mathsrevision.com Creation of a tan graph Graphs Let’s investigate created by Mr. Lafferty

Sine Graph www.mathsrevision.com Key Features Zeros (Root) at 0, 180o and 360o Max value occurs at x = 90o Nat 5 Mini value occurs at x = 270o Key Features www.mathsrevision.com (Period is every 360o) Maximum value of 1 - AMPLITUDE Minimum value of -1 created by Mr. Lafferty

Cosine Graphs www.mathsrevision.com Key Features Zeros (Roots) at 90o and 270o Max value occurs at x = 0o and 360o Nat 5 Minimum value occurs at x = 180o Key Features www.mathsrevision.com (Period is 360o) Maximum value of 1 - AMPLITUDE Minimum value of -1 created by Mr. Lafferty

Tangent Graphs www.mathsrevision.com Key Features Zeros (Roots) at 0 and 180o Nat 5 Key Features www.mathsrevision.com (Period is 180o) created by Mr. Lafferty

Trig Graphs Work through N5 TJ (Page 157) Ex 16.1 , 16.2 and 16.3 Nat 5 Work through N5 TJ Ex 16.1 , 16.2 and 16.3 (Page 157) www.mathsrevision.com created by Mr. Lafferty

Starter Nat 5 www.mathsrevision.com created by Mr. Lafferty

Sine & Cosine Graph www.mathsrevision.com Nat 5 Learning Intention Success Criteria To investigate graphs of the form y = a sin xo y = a cos xo Identify the key points for various trig graphs including Amplitude Period Roots. www.mathsrevision.com created by Mr. Lafferty

Sine Graph www.mathsrevision.com Key Features Zeros at 0, 180o and 360o Max value at x = 90o Nat 5 Minimum value at x = 270o Key Features www.mathsrevision.com (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

What effect does the number at the front have on the graphs ? y = sinxo y = 2sinxo y = 3sinxo y = 0.5sinxo y = -sinxo Sine Graph Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 Demo created by Mr. Lafferty

Sine Graph y = a sin (x) www.mathsrevision.com Nat 5 y = a sin (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For 0 < a < 1 compresses graph in the y - axis direction For a negative flips graph in the x – axis. created by Mr. Lafferty

Sine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 5sinxo y = 4sinxo Nat 5 6 4 2 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

Cosine Graphs www.mathsrevision.com Key Features Zeros at 90o and 270o Max value at x = 0o and 360o Nat 5 Minimum value at x = 180o Key Features www.mathsrevision.com (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty

What effect does the number at the front have on the graphs ? y = cosxo y = 2cosxo y = 3cosxo y = 0.5cosxo y = -cosxo Cosine Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 Demo created by Mr. Lafferty

Cosine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = cosxo y = 4cosxo Nat 5 6 4 2 www.mathsrevision.com 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty

Trig Graphs Now try N5 TJ (Page 161) Ex 16.4 www.mathsrevision.com Nat 5 Now try N5 TJ Ex 16.4 (Page 161) www.mathsrevision.com created by Mr. Lafferty

Starter Nat 5 www.mathsrevision.com created by Mr. Lafferty

Trig Graphs www.mathsrevision.com Nat 5 Learning Intention Success Criteria To investigate graphs of the form y = a sin bxo y = a cos bxo Identify the key points for various trig graphs including Amplitude Period Roots. www.mathsrevision.com created by Mr. Lafferty

Period of a Function y = sin bx www.mathsrevision.com Nat 5 When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360o www.mathsrevision.com Let’s investigate the function y = sin bx created by Mr. Lafferty

What effect does the number in front of x have on the graphs ? y = sinxo y = sin2xo y = sin4xo y = sin0.5xo Sine Graph Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 Demo created by Mr. Lafferty

Trigonometry Graphs y = a sin (bx) www.mathsrevision.com Nat 5 y = a sin (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For 0 < a < 1 compresses graph in the y - axis direction For a negative flips graph in the x – axis. created by Mr. Lafferty

What effect does the number at the front have on the graphs ? Cosine y = cosxo y = cos2xo y = cos3xo Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Trigonometry Graphs y = a cos (bx) www.mathsrevision.com Nat 5 y = a cos (bx) How many times it repeats itself in 360o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For 0 < a < 1 compresses graph in the y - axis direction For a negative flips graph in the x – axis. created by Mr. Lafferty

Trigonometry Graphs y = a tan (bx) www.mathsrevision.com Nat 5 y = a tan (bx) How many times it repeats itself in 180o www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For 0 < a < 1 compresses graph in the y - axis direction For a negative flips graph in the x – axis. created by Mr. Lafferty

Write down equations for graphs shown ? y = 0.5sin2xo y = 2sin4xo y = 3sin0.5xo Trig Graph Combinations Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 Demo created by Mr. Lafferty

Write down equations for the graphs shown? y = 1.5cos2xo y = -2cos2xo y = 0.5cos4xo Cosine Combinations Nat 5 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Trig Graphs Now try N5 TJ (Page 163) Ex 16.5 www.mathsrevision.com Nat 5 Now try N5 TJ Ex 16.5 (Page 163) www.mathsrevision.com created by Mr. Lafferty

Starter Nat 5 www.mathsrevision.com created by Mr. Lafferty

y = asinxo + b www.mathsrevision.com Nat 5 Learning Intention Success Criteria We are learning how to sketch graphs of the type y = asinxo + b y = acosxo + b Identify and sketch the key points for various trig graphs including Amplitude Period Roots. www.mathsrevision.com created by Mr. Lafferty

Write down equations for graphs shown ? y = 0.5sin2xo + 0.5 y = 2sin4xo- 1 Trig Graph Combinations Higher 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 Demo -3 created by Mr. Lafferty

Write down the equations for the graphs shown? Trig Graphs y = cos2xo + 1 y = -2cos2xo - 1 DEMO Combinations Higher 3 2 1 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

Trig Graphs Now try N5 TJ (Page 165) Ex 16.6 www.mathsrevision.com Nat 5 Now try N5 TJ Ex 16.6 (Page 165) www.mathsrevision.com created by Mr. Lafferty

Starter Nat 5 www.mathsrevision.com created by Mr. Lafferty

Phase Angle www.mathsrevision.com Nat 5 Learning Intention Success Criteria To investigate graphs of the form y = asin(xo + b) y = acos(xo + b) Identify and sketch the key points for trig graphs of the form y = asin(xo + b) y = acos(xo + b) www.mathsrevision.com created by Mr. Lafferty

Phase Angle y = sin(x + 60)o www.mathsrevision.com 1 To the left “+” By how much do we have to move the standard sine curve so it fits on the other sine curve? Phase Angle Nat 5 y = sin(x + 60)o 1 To the left “+” 60o www.mathsrevision.com -60o 90o 180o 270o 360o -1 created by Mr. Lafferty

Phase Angle y = sin(x - 45)o www.mathsrevision.com 1 To the right “-” By how much do we have to move the standard sine curve so it fits on the other sine curve? Phase Angle Nat 5 y = sin(x - 45)o 1 To the right “-” 45o www.mathsrevision.com 45o 90o 180o 270o 360o -1 Demo created by Mr. Lafferty

Phase Angle y = sin (x + b) www.mathsrevision.com Moves graph Nat 5 y = sin (x + b) Moves graph along x - axis www.mathsrevision.com For c > 0 moves graph to the left along x – axis For c < 0 moves graph to the right along x – axis created by Mr. Lafferty

Phase Angle y = cos(x - 70)o www.mathsrevision.com 1 To the right “-” By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Phase Angle Nat 5 y = cos(x - 70)o 1 To the right “-” 70o www.mathsrevision.com 90o 160o 180o 270o 360o -1 created by Mr. Lafferty

Phase Angle y = cos(x + 56)o www.mathsrevision.com 1 To the left “+” By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Phase Angle Nat 5 y = cos(x + 56)o 1 To the left “+” 56o www.mathsrevision.com 34o 90o 180o 270o 360o -1 created by Mr. Lafferty

y = a sin (x + b) Summary of work So far www.mathsrevision.com Nat 5 y = a sin (x + b) For a > 1 stretches graph in the y-axis direction For b > 0 moves graph to the left along x – axis For 0 < a < 1 compresses graph in the y - axis direction For b < 0 moves graph to the right along x – axis www.mathsrevision.com For a - negative flips graph in the x – axis. created by Mr. Lafferty

Phase Angle Now try N5 TJ (Page 168) Ex 16.7 www.mathsrevision.com Nat 5 Now try N5 TJ Ex 16.7 (Page 168) www.mathsrevision.com created by Mr. Lafferty