r=1+cos(x)

Rewrite the expression as cos(x)+1.

cos(x)+1

Use the form acos(bx-c)+d to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a=1

b=1

c=0

d=1

Find the amplitude |a|.

Amplitude: 1

The period of the function can be calculated using 2π|b|.

Period: 2π|b|

Replace b with 1 in the formula for period.

Period: 2π|1|

Solve the equation.

The absolute value is the distance between a number and zero. The distance between 0 and 1 is 1.

Period: 2π1

Divide 2π by 1.

Period: 2π

Period: 2π

Period: 2π

The phase shift of the function can be calculated from cb.

Phase Shift: cb

Replace the values of c and b in the equation for phase shift.

Phase Shift: 01

Divide 0 by 1.

Phase Shift: 0

Phase Shift: 0

Find the vertical shift d.

Vertical Shift: 1

List the properties of the trigonometric function.

Amplitude: 1

Period: 2π

Phase Shift: 0 (0 to the right)

Vertical Shift: 1

Find the point at x=0.

Replace the variable x with 0 in the expression.

f(0)=1+cos(0)

Simplify the result.

The exact value of cos(0) is 1.

f(0)=1+1

Add 1 and 1.

f(0)=2

The final answer is 2.

2

2

2

Find the point at x=π2.

Replace the variable x with π2 in the expression.

f(π2)=1+cos(π2)

Simplify the result.

The exact value of cos(π2) is 0.

f(π2)=1+0

Add 1 and 0.

f(π2)=1

The final answer is 1.

1

1

1

Find the point at x=π.

Replace the variable x with π in the expression.

f(π)=1+cos(π)

Simplify the result.

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

f(π)=1-cos(0)

The exact value of cos(0) is 1.

f(π)=1-1⋅1

Multiply -1 by 1.

f(π)=1-1

f(π)=1-1

Subtract 1 from 1.

f(π)=0

The final answer is 0.

0

0

0

Find the point at x=3π2.

Replace the variable x with 3π2 in the expression.

f(3π2)=1+cos(3π2)

Simplify the result.

Simplify each term.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

f(3π2)=1+cos(π2)

The exact value of cos(π2) is 0.

f(3π2)=1+0

f(3π2)=1+0

Add 1 and 0.

f(3π2)=1

The final answer is 1.

1

1

1

Find the point at x=2π.

Replace the variable x with 2π in the expression.

f(2π)=1+cos(2π)

Simplify the result.

Simplify each term.

2π is a full rotation so replace with 0.

f(2π)=1+cos(0)

The exact value of cos(0) is 1.

f(2π)=1+1

f(2π)=1+1

Add 1 and 1.

f(2π)=2

The final answer is 2.

2

2

2

List the points in a table.

xf(x)02π21π03π212π2

xf(x)02π21π03π212π2

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: 1

Period: 2π

Phase Shift: 0 (0 to the right)

Vertical Shift: 1

xf(x)02π21π03π212π2

Graph r=1+cos(x)