Precalculus Examples

$y = - 3 \cos (4 x + \frac{π}{3})$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 3 \cos (4 x + \frac{π}{3})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 3 \cos (4 x + \frac{π}{3}) = 0$ .

$- 3 \cos (4 x + \frac{π}{3}) = 0$

Step 1.2.2

Divide each term in $- 3 \cos (4 x + \frac{π}{3}) = 0$ by $- 3$ and simplify.

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Step 1.2.2.1

Divide each term in $- 3 \cos (4 x + \frac{π}{3}) = 0$ by $- 3$ .

$\frac{- 3 \cos (4 x + \frac{π}{3})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 \cos (4 x + \frac{π}{3})}{- 3} = \frac{0}{- 3}$

Step 1.2.2.2.1.2

Divide $\cos (4 x + \frac{π}{3})$ by $1$ .

$\cos (4 x + \frac{π}{3}) = \frac{0}{- 3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 3$ .

$\cos (4 x + \frac{π}{3}) = 0$

Step 1.2.3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$4 x + \frac{π}{3} = \arccos (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$4 x + \frac{π}{3} = \frac{π}{2}$

Step 1.2.5

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$4 x = \frac{π}{2} - \frac{π}{3}$

Step 1.2.5.2

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

$4 x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3}$

Step 1.2.5.3

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

$4 x = \frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.5.4

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

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Step 1.2.5.4.1

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

$4 x = \frac{π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.5.4.2

Multiply $2$ by $3$ .

$4 x = \frac{π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.5.4.3

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

$4 x = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$

Step 1.2.5.4.4

Multiply $3$ by $2$ .

$4 x = \frac{π \cdot 3}{6} - \frac{π \cdot 2}{6}$

Step 1.2.5.5

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 3 - π \cdot 2}{6}$

Step 1.2.5.6

Simplify the numerator.

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Step 1.2.5.6.1

Move $3$ to the left of $π$ .

$4 x = \frac{3 \cdot π - π \cdot 2}{6}$

Step 1.2.5.6.2

Multiply $2$ by $- 1$ .

$4 x = \frac{3 π - 2 π}{6}$

Step 1.2.5.6.3

Subtract $2 π$ from $3 π$ .

$4 x = \frac{π}{6}$

Step 1.2.6

Divide each term in $4 x = \frac{π}{6}$ by $4$ and simplify.

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Step 1.2.6.1

Divide each term in $4 x = \frac{π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 1.2.6.2

Simplify the left side.

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Step 1.2.6.2.1

Cancel the common factor of $4$ .

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Step 1.2.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{π}{6}}{4}$

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{4}$

Step 1.2.6.3

Simplify the right side.

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Step 1.2.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{4}$

Step 1.2.6.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{4}$ .

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Step 1.2.6.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{4}$ .

$x = \frac{π}{6 \cdot 4}$

Step 1.2.6.3.2.2

Multiply $6$ by $4$ .

$x = \frac{π}{24}$

Step 1.2.7

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$4 x + \frac{π}{3} = 2 π - \frac{π}{2}$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Simplify $2 π - \frac{π}{2}$ .

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Step 1.2.8.1.1

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

$4 x + \frac{π}{3} = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.8.1.2

Combine fractions.

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Step 1.2.8.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$4 x + \frac{π}{3} = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.8.1.2.2

Combine the numerators over the common denominator.

$4 x + \frac{π}{3} = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.8.1.3

Simplify the numerator.

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Step 1.2.8.1.3.1

Multiply $2$ by $2$ .

$4 x + \frac{π}{3} = \frac{4 π - π}{2}$

Step 1.2.8.1.3.2

Subtract $π$ from $4 π$ .

$4 x + \frac{π}{3} = \frac{3 π}{2}$

Step 1.2.8.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.8.2.1

Subtract $\frac{π}{3}$ from both sides of the equation.

$4 x = \frac{3 π}{2} - \frac{π}{3}$

Step 1.2.8.2.2

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

$4 x = \frac{3 π}{2} \cdot \frac{3}{3} - \frac{π}{3}$

Step 1.2.8.2.3

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

$4 x = \frac{3 π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.8.2.4

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

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Step 1.2.8.2.4.1

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

$4 x = \frac{3 π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.8.2.4.2

Multiply $2$ by $3$ .

$4 x = \frac{3 π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$

Step 1.2.8.2.4.3

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

$4 x = \frac{3 π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$

Step 1.2.8.2.4.4

Multiply $3$ by $2$ .

$4 x = \frac{3 π \cdot 3}{6} - \frac{π \cdot 2}{6}$

Step 1.2.8.2.5

Combine the numerators over the common denominator.

$4 x = \frac{3 π \cdot 3 - π \cdot 2}{6}$

Step 1.2.8.2.6

Simplify the numerator.

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Step 1.2.8.2.6.1

Multiply $3$ by $3$ .

$4 x = \frac{9 π - π \cdot 2}{6}$

Step 1.2.8.2.6.2

Multiply $2$ by $- 1$ .

$4 x = \frac{9 π - 2 π}{6}$

Step 1.2.8.2.6.3

Subtract $2 π$ from $9 π$ .

$4 x = \frac{7 π}{6}$

Step 1.2.8.3

Divide each term in $4 x = \frac{7 π}{6}$ by $4$ and simplify.

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Step 1.2.8.3.1

Divide each term in $4 x = \frac{7 π}{6}$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{7 π}{6}}{4}$

Step 1.2.8.3.2

Simplify the left side.

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Step 1.2.8.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.8.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{7 π}{6}}{4}$

Step 1.2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{7 π}{6}}{4}$

Step 1.2.8.3.3

Simplify the right side.

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Step 1.2.8.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{7 π}{6} \cdot \frac{1}{4}$

Step 1.2.8.3.3.2

Multiply $\frac{7 π}{6} \cdot \frac{1}{4}$ .

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Step 1.2.8.3.3.2.1

Multiply $\frac{7 π}{6}$ by $\frac{1}{4}$ .

$x = \frac{7 π}{6 \cdot 4}$

Step 1.2.8.3.3.2.2

Multiply $6$ by $4$ .

$x = \frac{7 π}{24}$

Step 1.2.9

Find the period of $\cos (4 x + \frac{π}{3})$ .

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Step 1.2.9.1

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

$\frac{2 π}{| b |}$

Step 1.2.9.2

Replace $b$ with $4$ in the formula for period.

$\frac{2 π}{| 4 |}$

Step 1.2.9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{2 π}{4}$

Step 1.2.9.4

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

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Step 1.2.9.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 1.2.9.4.2

Cancel the common factors.

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Step 1.2.9.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 1.2.9.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 1.2.9.4.2.3

Rewrite the expression.

$\frac{π}{2}$

Step 1.2.10

The period of the $\cos (4 x + \frac{π}{3})$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{24} + \frac{π n}{2}, \frac{7 π}{24} + \frac{π n}{2}$ , for any integer $n$

Step 1.2.11

Consolidate the answers.

$x = \frac{π}{24} + \frac{π n}{4}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{24} + \frac{π n}{4}, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 3 \cos (4 (0) + \frac{π}{3})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 3 \cos (4 (0) + \frac{π}{3})$

Step 2.2.2

Simplify $- 3 \cos (4 (0) + \frac{π}{3})$ .

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Step 2.2.2.1

Multiply $4$ by $0$ .

$y = - 3 \cos (0 + \frac{π}{3})$

Step 2.2.2.2

Add $0$ and $\frac{π}{3}$ .

$y = - 3 \cos (\frac{π}{3})$

Step 2.2.2.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$y = - 3 (\frac{1}{2})$

Step 2.2.2.4

Combine $- 3$ and $\frac{1}{2}$ .

$y = \frac{- 3}{2}$

Step 2.2.2.5

Move the negative in front of the fraction.

$y = - \frac{3}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{3}{2})$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{24} + \frac{π n}{4}, 0)$ , for any integer $n$

y-intercept(s): $(0, - \frac{3}{2})$

Step 4