Algebra Examples

$y = 10 \cos (\frac{2 π}{3} (x + \frac{1}{4}))$

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.

$a = 10$

$b = \frac{2 π}{3}$

$c = - \frac{π}{6}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $10$

Step 3

Find the period of $10 \cos (\frac{2 π x}{3} + \frac{π}{6})$ .

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Step 3.1

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

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{2 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{3} |}$

Step 3.3

$\frac{2 π}{3}$ is approximately $2.0943951$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{3}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2 π}$

Step 3.5

Cancel the common factor of $2 π$ .

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Step 3.5.1

Cancel the common factor.

$2 π \frac{3}{2 π}$

Step 3.5.2

Rewrite the expression.

$3$

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: $\frac{- \frac{π}{6}}{\frac{2 π}{3}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot \frac{3}{2 π}$

Step 4.4

Cancel the common factor of $π$ .

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Step 4.4.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot \frac{3}{2 π}$

Step 4.4.2

Factor $π$ out of $- π$ .

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{3}{2 π}$

Step 4.4.3

Factor $π$ out of $2 π$ .

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{3}{π \cdot 2}$

Step 4.4.4

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 1}{6} \cdot \frac{3}{π \cdot 2}$

Step 4.4.5

Rewrite the expression.

Phase Shift: $\frac{- 1}{6} \cdot \frac{3}{2}$

Step 4.5

Cancel the common factor of $3$ .

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Step 4.5.1

Factor $3$ out of $6$ .

Phase Shift: $\frac{- 1}{3 (2)} \cdot \frac{3}{2}$

Step 4.5.2

Cancel the common factor.

Phase Shift: $\frac{- 1}{3 \cdot 2} \cdot \frac{3}{2}$

Step 4.5.3

Rewrite the expression.

Phase Shift: $\frac{- 1}{2} \cdot \frac{1}{2}$

Step 4.6

Multiply $\frac{- 1}{2}$ by $\frac{1}{2}$ .

Phase Shift: $\frac{- 1}{2 \cdot 2}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Multiply $2$ by $2$ .

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

Step 4.7.2

Move the negative in front of the fraction.

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

Step 5

List the properties of the trigonometric function.

Amplitude: $10$

Period: $3$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = - \frac{1}{4}$ .

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Step 6.1.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

$f (- \frac{1}{4}) = 10 \cos (\frac{2 π (- \frac{1}{4})}{3} + \frac{π}{6})$

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

Simplify the numerator.

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Step 6.1.2.1.1.1

Multiply $- 1$ by $2$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{- 2 π \frac{1}{4}}{3} + \frac{π}{6})$

Step 6.1.2.1.1.2

Combine $\frac{1}{4}$ and $- 2$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{- 2}{4} \cdot π}{3} + \frac{π}{6})$

Step 6.1.2.1.1.3

Combine $\frac{- 2}{4}$ and $π$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{- 2 π}{4}}{3} + \frac{π}{6})$

Step 6.1.2.1.2

Simplify the numerator.

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Step 6.1.2.1.2.1

Reduce the expression $\frac{- 2 π}{4}$ by cancelling the common factors.

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Step 6.1.2.1.2.1.1

Factor $2$ out of $- 2 π$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{2 (- π)}{4}}{3} + \frac{π}{6})$

Step 6.1.2.1.2.1.2

Factor $2$ out of $4$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{2 (- π)}{2 (2)}}{3} + \frac{π}{6})$

Step 6.1.2.1.2.1.3

Cancel the common factor.

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{2 (- π)}{2 \cdot 2}}{3} + \frac{π}{6})$

Step 6.1.2.1.2.1.4

Rewrite the expression.

$f (- \frac{1}{4}) = 10 \cos (\frac{\frac{- π}{2}}{3} + \frac{π}{6})$

Step 6.1.2.1.2.2

Move the negative in front of the fraction.

$f (- \frac{1}{4}) = 10 \cos (\frac{- \frac{π}{2}}{3} + \frac{π}{6})$

Step 6.1.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{1}{4}) = 10 \cos (- \frac{π}{2} \cdot \frac{1}{3} + \frac{π}{6})$

Step 6.1.2.1.4

Multiply $- \frac{π}{2} \cdot \frac{1}{3}$ .

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Step 6.1.2.1.4.1

Multiply $\frac{1}{3}$ by $\frac{π}{2}$ .

$f (- \frac{1}{4}) = 10 \cos (- \frac{π}{3 \cdot 2} + \frac{π}{6})$

Step 6.1.2.1.4.2

Multiply $3$ by $2$ .

$f (- \frac{1}{4}) = 10 \cos (- \frac{π}{6} + \frac{π}{6})$

Step 6.1.2.2

Simplify terms.

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Step 6.1.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{1}{4}) = 10 \cos (\frac{- π + π}{6})$

Step 6.1.2.2.2

Add $- π$ and $π$ .

$f (- \frac{1}{4}) = 10 \cos (\frac{0}{6})$

Step 6.1.2.2.3

Divide $0$ by $6$ .

$f (- \frac{1}{4}) = 10 \cos (0)$

Step 6.1.2.3

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

$f (- \frac{1}{4}) = 10 \cdot 1$

Step 6.1.2.4

Multiply $10$ by $1$ .

$f (- \frac{1}{4}) = 10$

Step 6.1.2.5

The final answer is $10$ .

$10$

Step 6.2

Find the point at $x = \frac{1}{2}$ .

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Step 6.2.1

Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 10 \cos (\frac{2 π (\frac{1}{2})}{3} + \frac{π}{6})$

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

Simplify the numerator.

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Step 6.2.2.1.1.1

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{1}{2}) = 10 \cos (\frac{\frac{2}{2} \cdot π}{3} + \frac{π}{6})$

Step 6.2.2.1.1.2

Combine $\frac{2}{2}$ and $π$ .

$f (\frac{1}{2}) = 10 \cos (\frac{\frac{2 π}{2}}{3} + \frac{π}{6})$

Step 6.2.2.1.2

Reduce the expression by cancelling the common factors.

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Step 6.2.2.1.2.1

Reduce the expression $\frac{2 π}{2}$ by cancelling the common factors.

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Step 6.2.2.1.2.1.1

Cancel the common factor.

$f (\frac{1}{2}) = 10 \cos (\frac{\frac{2 π}{2}}{3} + \frac{π}{6})$

Step 6.2.2.1.2.1.2

Rewrite the expression.

$f (\frac{1}{2}) = 10 \cos (\frac{\frac{π}{1}}{3} + \frac{π}{6})$

Step 6.2.2.1.2.2

Divide $π$ by $1$ .

$f (\frac{1}{2}) = 10 \cos (\frac{π}{3} + \frac{π}{6})$

Step 6.2.2.2

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{1}{2}) = 10 \cos (\frac{π}{3} \cdot \frac{2}{2} + \frac{π}{6})$

Step 6.2.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.3.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$f (\frac{1}{2}) = 10 \cos (\frac{π \cdot 2}{3 \cdot 2} + \frac{π}{6})$

Step 6.2.2.3.2

Multiply $3$ by $2$ .

$f (\frac{1}{2}) = 10 \cos (\frac{π \cdot 2}{6} + \frac{π}{6})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = 10 \cos (\frac{π \cdot 2 + π}{6})$

Step 6.2.2.5

Simplify the numerator.

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Step 6.2.2.5.1

Move $2$ to the left of $π$ .

$f (\frac{1}{2}) = 10 \cos (\frac{2 \cdot π + π}{6})$

Step 6.2.2.5.2

Add $2 π$ and $π$ .

$f (\frac{1}{2}) = 10 \cos (\frac{3 π}{6})$

Step 6.2.2.6

Cancel the common factor of $3$ and $6$ .

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Step 6.2.2.6.1

Factor $3$ out of $3 π$ .

$f (\frac{1}{2}) = 10 \cos (\frac{3 (π)}{6})$

Step 6.2.2.6.2

Cancel the common factors.

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Step 6.2.2.6.2.1

Factor $3$ out of $6$ .

$f (\frac{1}{2}) = 10 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.2

Cancel the common factor.

$f (\frac{1}{2}) = 10 \cos (\frac{3 π}{3 \cdot 2})$

Step 6.2.2.6.2.3

Rewrite the expression.

$f (\frac{1}{2}) = 10 \cos (\frac{π}{2})$

Step 6.2.2.7

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

$f (\frac{1}{2}) = 10 \cdot 0$

Step 6.2.2.8

Multiply $10$ by $0$ .

$f (\frac{1}{2}) = 0$

Step 6.2.2.9

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{5}{4}$ .

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Step 6.3.1

Replace the variable $x$ with $\frac{5}{4}$ in the expression.

$f (\frac{5}{4}) = 10 \cos (\frac{2 π (\frac{5}{4})}{3} + \frac{π}{6})$

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

Simplify the numerator.

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Step 6.3.2.1.1.1

Combine $\frac{5}{4}$ and $2$ .

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{5 \cdot 2}{4} \cdot π}{3} + \frac{π}{6})$

Step 6.3.2.1.1.2

Combine $\frac{5 \cdot 2}{4}$ and $π$ .

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{5 \cdot (2 π)}{4}}{3} + \frac{π}{6})$

Step 6.3.2.1.2

Multiply $5$ by $2$ .

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{10 π}{4}}{3} + \frac{π}{6})$

Step 6.3.2.1.3

Reduce the expression $\frac{10 π}{4}$ by cancelling the common factors.

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Step 6.3.2.1.3.1

Factor $2$ out of $10 π$ .

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{2 (5 π)}{4}}{3} + \frac{π}{6})$

Step 6.3.2.1.3.2

Factor $2$ out of $4$ .

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{2 (5 π)}{2 (2)}}{3} + \frac{π}{6})$

Step 6.3.2.1.3.3

Cancel the common factor.

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{2 (5 π)}{2 \cdot 2}}{3} + \frac{π}{6})$

Step 6.3.2.1.3.4

Rewrite the expression.

$f (\frac{5}{4}) = 10 \cos (\frac{\frac{5 π}{2}}{3} + \frac{π}{6})$

Step 6.3.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5}{4}) = 10 \cos (\frac{5 π}{2} \cdot \frac{1}{3} + \frac{π}{6})$

Step 6.3.2.1.5

Multiply $\frac{5 π}{2} \cdot \frac{1}{3}$ .

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Step 6.3.2.1.5.1

Multiply $\frac{5 π}{2}$ by $\frac{1}{3}$ .

$f (\frac{5}{4}) = 10 \cos (\frac{5 π}{2 \cdot 3} + \frac{π}{6})$

Step 6.3.2.1.5.2

Multiply $2$ by $3$ .

$f (\frac{5}{4}) = 10 \cos (\frac{5 π}{6} + \frac{π}{6})$

Step 6.3.2.2

Simplify terms.

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Step 6.3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{5}{4}) = 10 \cos (\frac{5 π + π}{6})$

Step 6.3.2.2.2

Add $5 π$ and $π$ .

$f (\frac{5}{4}) = 10 \cos (\frac{6 π}{6})$

Step 6.3.2.2.3

Cancel the common factor of $6$ .

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Step 6.3.2.2.3.1

Cancel the common factor.

$f (\frac{5}{4}) = 10 \cos (\frac{6 π}{6})$

Step 6.3.2.2.3.2

Divide $π$ by $1$ .

$f (\frac{5}{4}) = 10 \cos (π)$

Step 6.3.2.3

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.

$f (\frac{5}{4}) = 10 (- \cos (0))$

Step 6.3.2.4

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

$f (\frac{5}{4}) = 10 (- 1 \cdot 1)$

Step 6.3.2.5

Multiply $10 (- 1 \cdot 1)$ .

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Step 6.3.2.5.1

Multiply $- 1$ by $1$ .

$f (\frac{5}{4}) = 10 \cdot - 1$

Step 6.3.2.5.2

Multiply $10$ by $- 1$ .

$f (\frac{5}{4}) = - 10$

Step 6.3.2.6

The final answer is $- 10$ .

$- 10$

Step 6.4

Find the point at $x = 2$ .

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Step 6.4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 10 \cos (\frac{2 π (2)}{3} + \frac{π}{6})$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Multiply $2$ by $2$ .

$f (2) = 10 \cos (\frac{4 π}{3} + \frac{π}{6})$

Step 6.4.2.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (2) = 10 \cos (\frac{4 π}{3} \cdot \frac{2}{2} + \frac{π}{6})$

Step 6.4.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.2.3.1

Multiply $\frac{4 π}{3}$ by $\frac{2}{2}$ .

$f (2) = 10 \cos (\frac{4 π \cdot 2}{3 \cdot 2} + \frac{π}{6})$

Step 6.4.2.3.2

Multiply $3$ by $2$ .

$f (2) = 10 \cos (\frac{4 π \cdot 2}{6} + \frac{π}{6})$

Step 6.4.2.4

Combine the numerators over the common denominator.

$f (2) = 10 \cos (\frac{4 π \cdot 2 + π}{6})$

Step 6.4.2.5

Simplify the numerator.

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Step 6.4.2.5.1

Multiply $2$ by $4$ .

$f (2) = 10 \cos (\frac{8 π + π}{6})$

Step 6.4.2.5.2

Add $8 π$ and $π$ .

$f (2) = 10 \cos (\frac{9 π}{6})$

Step 6.4.2.6

Cancel the common factor of $9$ and $6$ .

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Step 6.4.2.6.1

Factor $3$ out of $9 π$ .

$f (2) = 10 \cos (\frac{3 (3 π)}{6})$

Step 6.4.2.6.2

Cancel the common factors.

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Step 6.4.2.6.2.1

Factor $3$ out of $6$ .

$f (2) = 10 \cos (\frac{3 (3 π)}{3 (2)})$

Step 6.4.2.6.2.2

Cancel the common factor.

$f (2) = 10 \cos (\frac{3 (3 π)}{3 \cdot 2})$

Step 6.4.2.6.2.3

Rewrite the expression.

$f (2) = 10 \cos (\frac{3 π}{2})$

Step 6.4.2.7

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

$f (2) = 10 \cos (\frac{π}{2})$

Step 6.4.2.8

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

$f (2) = 10 \cdot 0$

Step 6.4.2.9

Multiply $10$ by $0$ .

$f (2) = 0$

Step 6.4.2.10

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{11}{4}$ .

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Step 6.5.1

Replace the variable $x$ with $\frac{11}{4}$ in the expression.

$f (\frac{11}{4}) = 10 \cos (\frac{2 π (\frac{11}{4})}{3} + \frac{π}{6})$

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

Simplify the numerator.

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Step 6.5.2.1.1.1

Combine $\frac{11}{4}$ and $2$ .

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{11 \cdot 2}{4} \cdot π}{3} + \frac{π}{6})$

Step 6.5.2.1.1.2

Combine $\frac{11 \cdot 2}{4}$ and $π$ .

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{11 \cdot (2 π)}{4}}{3} + \frac{π}{6})$

Step 6.5.2.1.2

Multiply $11$ by $2$ .

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{22 π}{4}}{3} + \frac{π}{6})$

Step 6.5.2.1.3

Reduce the expression $\frac{22 π}{4}$ by cancelling the common factors.

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Step 6.5.2.1.3.1

Factor $2$ out of $22 π$ .

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{2 (11 π)}{4}}{3} + \frac{π}{6})$

Step 6.5.2.1.3.2

Factor $2$ out of $4$ .

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{2 (11 π)}{2 (2)}}{3} + \frac{π}{6})$

Step 6.5.2.1.3.3

Cancel the common factor.

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{2 (11 π)}{2 \cdot 2}}{3} + \frac{π}{6})$

Step 6.5.2.1.3.4

Rewrite the expression.

$f (\frac{11}{4}) = 10 \cos (\frac{\frac{11 π}{2}}{3} + \frac{π}{6})$

Step 6.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{11}{4}) = 10 \cos (\frac{11 π}{2} \cdot \frac{1}{3} + \frac{π}{6})$

Step 6.5.2.1.5

Multiply $\frac{11 π}{2} \cdot \frac{1}{3}$ .

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Step 6.5.2.1.5.1

Multiply $\frac{11 π}{2}$ by $\frac{1}{3}$ .

$f (\frac{11}{4}) = 10 \cos (\frac{11 π}{2 \cdot 3} + \frac{π}{6})$

Step 6.5.2.1.5.2

Multiply $2$ by $3$ .

$f (\frac{11}{4}) = 10 \cos (\frac{11 π}{6} + \frac{π}{6})$

Step 6.5.2.2

Simplify terms.

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Step 6.5.2.2.1

Combine the numerators over the common denominator.

$f (\frac{11}{4}) = 10 \cos (\frac{11 π + π}{6})$

Step 6.5.2.2.2

Add $11 π$ and $π$ .

$f (\frac{11}{4}) = 10 \cos (\frac{12 π}{6})$

Step 6.5.2.2.3

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

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Step 6.5.2.2.3.1

Factor $6$ out of $12 π$ .

$f (\frac{11}{4}) = 10 \cos (\frac{6 (2 π)}{6})$

Step 6.5.2.2.3.2

Cancel the common factors.

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Step 6.5.2.2.3.2.1

Factor $6$ out of $6$ .

$f (\frac{11}{4}) = 10 \cos (\frac{6 (2 π)}{6 (1)})$

Step 6.5.2.2.3.2.2

Cancel the common factor.

$f (\frac{11}{4}) = 10 \cos (\frac{6 (2 π)}{6 \cdot 1})$

Step 6.5.2.2.3.2.3

Rewrite the expression.

$f (\frac{11}{4}) = 10 \cos (\frac{2 π}{1})$

Step 6.5.2.2.3.2.4

Divide $2 π$ by $1$ .

$f (\frac{11}{4}) = 10 \cos (2 π)$

Step 6.5.2.3

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

$f (\frac{11}{4}) = 10 \cos (0)$

Step 6.5.2.4

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

$f (\frac{11}{4}) = 10 \cdot 1$

Step 6.5.2.5

Multiply $10$ by $1$ .

$f (\frac{11}{4}) = 10$

Step 6.5.2.6

The final answer is $10$ .

$10$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{1}{4} & 10 \\ \frac{1}{2} & 0 \\ \frac{5}{4} & - 10 \\ 2 & 0 \\ \frac{11}{4} & 10 \end{array}$

Step 7

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

Amplitude: $10$

Period: $3$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{1}{4} & 10 \\ \frac{1}{2} & 0 \\ \frac{5}{4} & - 10 \\ 2 & 0 \\ \frac{11}{4} & 10 \end{array}$

Step 8