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Graph of Sine function example 1

Draw Sine function example 1

Question
Draw the graph of y = 1 + sin(x - pi/2)
Solution:
Step 1. Draw the graph of y = sin x
draw the graph of y = sin x from zero to 2pi
Step 2. Move the graph one unit up because it is y = 1 + sin(x - pi/2)
draw the graph of y = 1 + sin x
Step 3. Move the graph toward right direction pi/2 units because it is y = sin(x - pi/2) + 1
draw the graph of y = sin(x - pi/2) + 1
draw the graph of y = sin(x - pi/2) + 1 and y = sin x + 1 and y = sic x.

Sine function example 2

Question:
In the interval x >= 0 and x <= pi, what is the value of x when sin(6x) passes the x-axis the fifth time?
Solution
For y = sin(6x), B = 6, period = 2pi/B = 2pi/6 = pi/3. So the graph of y = sin(6x) passes the x-axis at 0, pi/6, pi/3, pi/2, 2pi/3, 5pi/6, pi. So that, in the interval [0, pi], x = 5pi/6 is the fifth time when the curve passes the x-axis.
draw the graph of y = sin 6x and find the value of x when y pass zero.

Function example 3

Question:
If f(x) = 3x2 - 2x + 1, what value of x will make f(x) a minimum?
Solution:
f(x) = 3x2 - 2x + 1
= 3[x2 - (2/3)x + 1/3]
= 3[x2 - (2/3)x + (1/3)2 - (1/3)2 + 1/3]
= 3[x2 - (2/3)x + 1/9 - 1/9 + 1/3]
= 3[x2 - (2/3)x + 1/9 - 1/9 + 3/9]
= 3[x2 - (2/3)x + 1/9 + 2/9]
= 3[(x - 1/3)2 + 2/9]
since (x - 1/3)2 >= 0, so the smallest value of (x - 1/3)2 is 0
thus, x = 1/3 make f(x) a minimum.
so when x = 1/3, f(x) has the minimum value, and f(1/3) = 3 × 2/9 = 6/9 = 2/3.