Algebra Examples

$y = \cos (\frac{θ}{4} + \frac{π}{4}) - 2$

Step 1

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

$a = 1$

$b = \frac{1}{4}$

$c = - \frac{π}{4}$

$d = - 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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Step 3.1

Find the period of $\cos (\frac{x}{4} + \frac{π}{4})$ .

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Step 3.1.1

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

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.1.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 3.1.5

Multiply $4$ by $2$ .

$8 π$

Step 3.2

Find the period of $- 2$ .

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Step 3.2.1

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

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.2.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 3.2.5

Multiply $4$ by $2$ .

$8 π$

Step 3.3

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

$8 π$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

Phase Shift: $\frac{- \frac{π}{4}}{\frac{1}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{4} \cdot 4$

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

Phase Shift: $\frac{- π}{4} \cdot 4$

Step 4.4.2

Cancel the common factor.

Phase Shift: $\frac{- π}{4} \cdot 4$

Step 4.4.3

Rewrite the expression.

Phase Shift: $- π$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $8 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: $- 2$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - π$ .

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Step 6.1.1

Replace the variable $x$ with $- π$ in the expression.

$f (- π) = \cos (\frac{- π}{4} + \frac{π}{4}) - 2$

Step 6.1.2

Simplify the result.

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Step 6.1.2.1

Simplify each term.

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Step 6.1.2.1.1

Combine the numerators over the common denominator.

$f (- π) = \cos (\frac{- π + π}{4}) - 2$

Step 6.1.2.1.2

Add $- π$ and $π$ .

$f (- π) = \cos (\frac{0}{4}) - 2$

Step 6.1.2.1.3

Divide $0$ by $4$ .

$f (- π) = \cos (0) - 2$

Step 6.1.2.1.4

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

$f (- π) = 1 - 2$

Step 6.1.2.2

Subtract $2$ from $1$ .

$f (- π) = - 1$

Step 6.1.2.3

The final answer is $- 1$ .

$- 1$

Step 6.2

Find the point at $x = π$ .

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Step 6.2.1

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

$f (π) = \cos (\frac{π}{4} + \frac{π}{4}) - 2$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Combine the numerators over the common denominator.

$f (π) = \cos (\frac{π + π}{4}) - 2$

Step 6.2.2.1.2

Add $π$ and $π$ .

$f (π) = \cos (\frac{2 π}{4}) - 2$

Step 6.2.2.1.3

Cancel the common factor of $2$ and $4$ .

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Step 6.2.2.1.3.1

Factor $2$ out of $2 π$ .

$f (π) = \cos (\frac{2 (π)}{4}) - 2$

Step 6.2.2.1.3.2

Cancel the common factors.

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Step 6.2.2.1.3.2.1

Factor $2$ out of $4$ .

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

Step 6.2.2.1.3.2.2

Cancel the common factor.

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

Step 6.2.2.1.3.2.3

Rewrite the expression.

$f (π) = \cos (\frac{π}{2}) - 2$

Step 6.2.2.1.4

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

$f (π) = 0 - 2$

Step 6.2.2.2

Subtract $2$ from $0$ .

$f (π) = - 2$

Step 6.2.2.3

The final answer is $- 2$ .

$- 2$

Step 6.3

Find the point at $x = 3 π$ .

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Step 6.3.1

Replace the variable $x$ with $3 π$ in the expression.

$f (3 π) = \cos (\frac{3 π}{4} + \frac{π}{4}) - 2$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Combine the numerators over the common denominator.

$f (3 π) = \cos (\frac{3 π + π}{4}) - 2$

Step 6.3.2.1.2

Add $3 π$ and $π$ .

$f (3 π) = \cos (\frac{4 π}{4}) - 2$

Step 6.3.2.1.3

Cancel the common factor of $4$ .

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Step 6.3.2.1.3.1

Cancel the common factor.

$f (3 π) = \cos (\frac{4 π}{4}) - 2$

Step 6.3.2.1.3.2

Divide $π$ by $1$ .

$f (3 π) = \cos (π) - 2$

Step 6.3.2.1.4

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.

$f (3 π) = - \cos (0) - 2$

Step 6.3.2.1.5

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

$f (3 π) = - 1 \cdot 1 - 2$

Step 6.3.2.1.6

Multiply $- 1$ by $1$ .

$f (3 π) = - 1 - 2$

Step 6.3.2.2

Subtract $2$ from $- 1$ .

$f (3 π) = - 3$

Step 6.3.2.3

The final answer is $- 3$ .

$- 3$

Step 6.4

Find the point at $x = 5 π$ .

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Step 6.4.1

Replace the variable $x$ with $5 π$ in the expression.

$f (5 π) = \cos (\frac{5 π}{4} + \frac{π}{4}) - 2$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Combine the numerators over the common denominator.

$f (5 π) = \cos (\frac{5 π + π}{4}) - 2$

Step 6.4.2.1.2

Add $5 π$ and $π$ .

$f (5 π) = \cos (\frac{6 π}{4}) - 2$

Step 6.4.2.1.3

Cancel the common factor of $6$ and $4$ .

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Step 6.4.2.1.3.1

Factor $2$ out of $6 π$ .

$f (5 π) = \cos (\frac{2 (3 π)}{4}) - 2$

Step 6.4.2.1.3.2

Cancel the common factors.

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Step 6.4.2.1.3.2.1

Factor $2$ out of $4$ .

$f (5 π) = \cos (\frac{2 (3 π)}{2 (2)}) - 2$

Step 6.4.2.1.3.2.2

Cancel the common factor.

$f (5 π) = \cos (\frac{2 (3 π)}{2 \cdot 2}) - 2$

Step 6.4.2.1.3.2.3

Rewrite the expression.

$f (5 π) = \cos (\frac{3 π}{2}) - 2$

Step 6.4.2.1.4

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

$f (5 π) = \cos (\frac{π}{2}) - 2$

Step 6.4.2.1.5

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

$f (5 π) = 0 - 2$

Step 6.4.2.2

Subtract $2$ from $0$ .

$f (5 π) = - 2$

Step 6.4.2.3

The final answer is $- 2$ .

$- 2$

Step 6.5

Find the point at $x = 7 π$ .

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Step 6.5.1

Replace the variable $x$ with $7 π$ in the expression.

$f (7 π) = \cos (\frac{7 π}{4} + \frac{π}{4}) - 2$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Simplify each term.

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Step 6.5.2.1.1

Combine the numerators over the common denominator.

$f (7 π) = \cos (\frac{7 π + π}{4}) - 2$

Step 6.5.2.1.2

Add $7 π$ and $π$ .

$f (7 π) = \cos (\frac{8 π}{4}) - 2$

Step 6.5.2.1.3

Cancel the common factor of $8$ and $4$ .

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Step 6.5.2.1.3.1

Factor $4$ out of $8 π$ .

$f (7 π) = \cos (\frac{4 (2 π)}{4}) - 2$

Step 6.5.2.1.3.2

Cancel the common factors.

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Step 6.5.2.1.3.2.1

Factor $4$ out of $4$ .

$f (7 π) = \cos (\frac{4 (2 π)}{4 (1)}) - 2$

Step 6.5.2.1.3.2.2

Cancel the common factor.

$f (7 π) = \cos (\frac{4 (2 π)}{4 \cdot 1}) - 2$

Step 6.5.2.1.3.2.3

Rewrite the expression.

$f (7 π) = \cos (\frac{2 π}{1}) - 2$

Step 6.5.2.1.3.2.4

Divide $2 π$ by $1$ .

$f (7 π) = \cos (2 π) - 2$

Step 6.5.2.1.4

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

$f (7 π) = \cos (0) - 2$

Step 6.5.2.1.5

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

$f (7 π) = 1 - 2$

Step 6.5.2.2

Subtract $2$ from $1$ .

$f (7 π) = - 1$

Step 6.5.2.3

The final answer is $- 1$ .

$- 1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - π & - 1 \\ π & - 2 \\ 3 π & - 3 \\ 5 π & - 2 \\ 7 π & - 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $8 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: $- 2$

$\begin{array}{c} x & f (x) \\ - π & - 1 \\ π & - 2 \\ 3 π & - 3 \\ 5 π & - 2 \\ 7 π & - 1 \end{array}$

Step 8