1. Graphs of Trigonometric Functions

2. Symmetry with respect to a point
Symmetry with respect to the axis or line A graph is said to be symmetric with respect to a line if the reflection (mirror image) about the line of every point on the graph is also on the graph The line is known as the line of symmetry.
Symmetry with respect to a point
A graph is said to be symmetric with respect to a point Q if to each point P on the graph, we can find point P’ on the same graph, such that Q is the midpoint of the segment joining P and P’.
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3. Two points are symmetric with respect to the y – axis if and only if their x – coordinates are additive inverses and they have the same y – coordinate.
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4. Two points are symmetric with respect to the x – axis if and only if their y –coordinates are additive inverses and they have the same x – coordinate.
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5. Two points are symmetric with respect to the origin if and only if both their x – and y – coordinates are additive inverses of each other.
Imagine sticking a pin in the given figure at the origin and then rotating the figure at Points P and P1 would be interchanged. The entire figure would look exactly as it did before rotating.
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6. A function is an even function when f(-x) = f(x) for all x in the domain of f. This is a function symmetric with respect to the y – axis.
A function is an odd function when f(-x) = - f(x) for all x in the domain of f. This is a function symmetric with respect to the origin.
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7. Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. -1 1
sin x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x
y = sin x
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8. Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1 -1
cos x x
Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.
y = cos x y x
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Cosine Function

9. Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an x-interval of
6. The cycle repeats itself indefinitely in both directions of
the x-axis.
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10. Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.
max
x-int
min 3 -3
y = 3 cos x 2  x y x
(0, 3)
( , 3)
( , 0)
( , 0)
( , –3)
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Example: y = 3 cos x

11. If |a| > 1, the amplitude stretches the graph vertically.
The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| < 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis. y x
y = 2 sin x
y = sin x
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
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Amplitude

12. If b > 1, the graph of the function is shrunk horizontally.
The period of a function is the x interval needed for the function to complete one cycle.
For b  0, the period of y = a sin bx is
For b  0, the period of y = a cos bx is also
If b > 1, the graph of the function is shrunk horizontally. y x
period: 2
period:
If 0 < b < 1, the graph of the function is stretched horizontally. y x
period: 4
period: 2
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Period of a Function

13. Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x
y = sin (–x)
Use the identity sin (–x) = – sin x
y = sin x
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x. y x
Use the identity cos (–x) = cos x
y = cos (–x)
y = cos (–x)
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Graph y = f(-x)

14. Identify the amplitude = .
Steps in Graphing y = a sin bx and y = a cos bx.
Identify the amplitude =
2. Find the period =
3. Find the intervals.
4. Apply the pattern, then graph.
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15. y = a cos bx
y = a sin bx
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16. Use the identity sin (– x) = – sin x:
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
Use the identity sin (– x) = – sin x:
y = 2 sin (–3x) = –2 sin 3x
period: 2 3 =
amplitude: |a| = |–2| = 2
Calculate the five key points. 2 –2
y = –2 sin 3x x y x
( , 2)
(0, 0)
( , 0)
( , 0)
( , -2)
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Example: y = 2 sin(-3x)

17. Graph of the Tangent Function
To graph y = tan x, use the identity
At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y x
Properties of y = tan x
1. domain : all real x
2. range: (–, +)
3. period: 
4. vertical asymptotes:
period:
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Tangent Function

18. Steps in Graphing y = a tan bx.
1. Determine the period
2. Locate two adjacent vertical asymptotes by solving for x:
3. Sketch the two vertical asymptotes found in Step 2.
4. Divide the interval into four equal parts.
5. Evaluate the function for the first – quarter point, midpoint, and third - quarter point, using the x – values in Step 4.
6. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.
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19. Example: Find the period and asymptotes and sketch the graph of
1. Period of y = tan x is  .
2. Find consecutive vertical
asymptotes by solving for x:
Vertical asymptotes:
3. Plot several points in
4. Sketch one branch and repeat.
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20. Example: Find the period and asymptotes and sketch the graph of
1. Period of y = tan x is of
Period is
2. Find consecutive vertical
asymptotes by solving for x:
Vertical asymptotes:
3. Divide - to  into four equal
parts.
4. Sketch one branch and repeat.
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21. 2. Vertical asymptotes are
x = - 2
x = 2 y x
Graph
1. Period is or 4.
2. Vertical asymptotes are
3. Divide the interval - 2 to 2 into four equal parts.
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22. Reporters: Naco, Sheree Pascua, Nathaniel Patacsil, Demi Peña, Kevin Peñaflor, Jezer