Graph y=cos(pi/12x)+6

$y = \cos (\frac{π}{12} x) + 6$

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = \frac{π}{12}$

$c = 0$

$d = 6$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period using the formula $\frac{2 π}{| b |}$ .

Find the period of $\cos (\frac{π x}{12})$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{12}$ in the formula for period.

$\frac{2 π}{| \frac{π}{12} |}$

$\frac{π}{12}$ is approximately $0.26179938$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{12}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{12}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{12}{π}$

Cancel the common factor.

$π \cdot 2 \frac{12}{π}$

Rewrite the expression.

$2 \cdot 12$

Multiply $2$ by $12$ .

$24$

Find the period of $6$ .

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{π}{12}$ in the formula for period.

$\frac{2 π}{| \frac{π}{12} |}$

$\frac{π}{12}$ is approximately $0.26179938$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{12}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{12}{π}$

Cancel the common factor of $π$ .

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{12}{π}$

Cancel the common factor.

$π \cdot 2 \frac{12}{π}$

Rewrite the expression.

$2 \cdot 12$

Multiply $2$ by $12$ .

$24$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$24$

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

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

Phase Shift: $\frac{0}{\frac{π}{12}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{12}{π})$

Multiply $0$ by $\frac{12}{π}$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $24$

Phase Shift: None

Vertical Shift: $6$

Select a few points to graph.

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \cos (\frac{π (0)}{12}) + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of $0$ and $12$ .

Factor $12$ out of $π (0)$ .

$f (0) = \cos (\frac{12 (π \cdot (0))}{12}) + 6$

Cancel the common factors.

Factor $12$ out of $12$ .

$f (0) = \cos (\frac{12 (π \cdot (0))}{12 (1)}) + 6$

Cancel the common factor.

$f (0) = \cos (\frac{12 (π \cdot (0))}{12 \cdot 1}) + 6$

Rewrite the expression.

$f (0) = \cos (\frac{π \cdot (0)}{1}) + 6$

Divide $π \cdot (0)$ by $1$ .

$f (0) = \cos (π \cdot (0)) + 6$

Multiply $π$ by $0$ .

$f (0) = \cos (0) + 6$

The exact value of $\cos (0)$ is $1$ .

$f (0) = 1 + 6$

Add $1$ and $6$ .

$f (0) = 7$

The final answer is $7$ .

$7$

Find the point at $x = 6$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = \cos (\frac{π (6)}{12}) + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of $6$ and $12$ .

Factor $6$ out of $π (6)$ .

$f (6) = \cos (\frac{6 π}{12}) + 6$

Cancel the common factors.

Factor $6$ out of $12$ .

$f (6) = \cos (\frac{6 π}{6 \cdot 2}) + 6$

Cancel the common factor.

$f (6) = \cos (\frac{6 π}{6 \cdot 2}) + 6$

Rewrite the expression.

$f (6) = \cos (\frac{π}{2}) + 6$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (6) = 0 + 6$

Add $0$ and $6$ .

$f (6) = 6$

The final answer is $6$ .

$6$

Find the point at $x = 12$ .

Replace the variable $x$ with $12$ in the expression.

$f (12) = \cos (\frac{π (12)}{12}) + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of $12$ .

Cancel the common factor.

$f (12) = \cos (\frac{π \cdot 12}{12}) + 6$

Divide $π$ by $1$ .

$f (12) = \cos (π) + 6$

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.

$f (12) = - \cos (0) + 6$

The exact value of $\cos (0)$ is $1$ .

$f (12) = - 1 \cdot 1 + 6$

Multiply $- 1$ by $1$ .

$f (12) = - 1 + 6$

Add $- 1$ and $6$ .

$f (12) = 5$

The final answer is $5$ .

$5$

Find the point at $x = 18$ .

Replace the variable $x$ with $18$ in the expression.

$f (18) = \cos (\frac{π (18)}{12}) + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of $18$ and $12$ .

Factor $6$ out of $π (18)$ .

$f (18) = \cos (\frac{6 (π \cdot (3))}{12}) + 6$

Cancel the common factors.

Factor $6$ out of $12$ .

$f (18) = \cos (\frac{6 (π \cdot (3))}{6 (2)}) + 6$

Cancel the common factor.

$f (18) = \cos (\frac{6 (π \cdot (3))}{6 \cdot 2}) + 6$

Rewrite the expression.

$f (18) = \cos (\frac{π \cdot (3)}{2}) + 6$

Move $3$ to the left of $π$ .

$f (18) = \cos (\frac{3 \cdot π}{2}) + 6$

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

$f (18) = \cos (\frac{π}{2}) + 6$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (18) = 0 + 6$

Add $0$ and $6$ .

$f (18) = 6$

The final answer is $6$ .

$6$

Find the point at $x = 24$ .

Replace the variable $x$ with $24$ in the expression.

$f (24) = \cos (\frac{π (24)}{12}) + 6$

Simplify the result.

Simplify each term.

Cancel the common factor of $24$ and $12$ .

Factor $12$ out of $π (24)$ .

$f (24) = \cos (\frac{12 (π \cdot (2))}{12}) + 6$

Cancel the common factors.

Factor $12$ out of $12$ .

$f (24) = \cos (\frac{12 (π \cdot (2))}{12 (1)}) + 6$

Cancel the common factor.

$f (24) = \cos (\frac{12 (π \cdot (2))}{12 \cdot 1}) + 6$

Rewrite the expression.

$f (24) = \cos (\frac{π \cdot (2)}{1}) + 6$

Divide $π \cdot (2)$ by $1$ .

$f (24) = \cos (π \cdot (2)) + 6$

Move $2$ to the left of $π$ .

$f (24) = \cos (2 \cdot π) + 6$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (24) = \cos (0) + 6$

The exact value of $\cos (0)$ is $1$ .

$f (24) = 1 + 6$

Add $1$ and $6$ .

$f (24) = 7$

The final answer is $7$ .

$7$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 7 \\ 6 & 6 \\ 12 & 5 \\ 18 & 6 \\ 24 & 7 \end{array}$

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

Amplitude: $1$

Period: $24$

Phase Shift: None

Vertical Shift: $6$

$\begin{array}{c} x & f (x) \\ 0 & 7 \\ 6 & 6 \\ 12 & 5 \\ 18 & 6 \\ 24 & 7 \end{array}$

Graph y=cos(pi/12x)+6

Related Questions

Support LCMGCF.com by disabling your adblocker.