Answer to Question #157767 in Calculus for Jon jay mendoza

Question

Show complete solution on the upper half of the graphing paper and the graph on the lower half portion. One function is the one graphing paper.

1.f(x)=2 sin (x-​π/​4)

2.f(x)=sin (x+π/​3)

3.f(x)=3 cos (x+π/​6)

4.f(x)=2 cos (x-2π)

5.f(x)=tan x

Answer 1

Q1.

f(x) = 2sin(x - π/4)

=> f(x)= 2(sinx cosπ/4 - cosx sinπ/4)

=> f(x) = 2((1/$\sqrt{2}$ ) sinx - (1/$\sqrt{2}$ )cosx)

=> f(x) = $\sqrt{2}$ (sinx - cosx)

Graph of y = sinx will be shifted right by π/4 and then it will be stretched twice vertically

Q2.

f(x) = sin(x + π/3)

=> f(x)= (sinx cosπ/3 + cosx sinπ/3)

=> f(x) = ((1/2) sinx + ($\sqrt{3}$ /2)cosx)

=> f(x) = (1/2)(sinx +$\sqrt{3}$ cosx)

Graph of y = sinx will be shifted left by π/3

Q3.

f(x) = 3cos(x + π/6)

f(x) = 3{ cosx cosπ/6 - sinx sin π/6}

=> f(x) = 3{($\sqrt{3}$ /2)cosx - (1/2)sinx}

=> f(x) = (3/2)($\sqrt{3}$ cosx - sinx)

Graph of y = cosx will be shifted left by π/6 and then it will be stretched three times vertically

Q4.

f(x) = 2cos(x - 2π)

=> f(x) = 2cos(2π - x)

=> f(x) = 2cosx

Graph of y = cosx will be stretched twice vertically.

Q5.

f(x) = tanx

f(x) = tanx is a discontinuous function at x = (2n+1)(π/2), n is integer. tanx is increasing function from -∞ to ∞ as x changes from -π/2 to π/2 . Similarly in the other slots like (-5π/2, -3π/2),(-3π/2, -π/2) (π/2, 3π/2) , (3π/2, 5π/2) etc.