Precalculus Examples

$y = - 5 \sin (4 x + \frac{π}{3})$

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 $x$ .

$0 = - 5 \sin (4 x + \frac{π}{3})$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 5 \sin (4 x + \frac{π}{3}) = 0$ .

$- 5 \sin (4 x + \frac{π}{3}) = 0$

Step 1.2.2

Divide each term in $- 5 \sin (4 x + \frac{π}{3}) = 0$ by $- 5$ and simplify.

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Step 1.2.2.1

Divide each term in $- 5 \sin (4 x + \frac{π}{3}) = 0$ by $- 5$ .

$\frac{- 5 \sin (4 x + \frac{π}{3})}{- 5} = \frac{0}{- 5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 5 \sin (4 x + \frac{π}{3})}{- 5} = \frac{0}{- 5}$

Step 1.2.2.2.1.2

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

$\sin (4 x + \frac{π}{3}) = \frac{0}{- 5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 5$ .

$\sin (4 x + \frac{π}{3}) = 0$

Step 1.2.3

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

$4 x + \frac{π}{3} = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

The exact value of $\arcsin (0)$ is $0$ .

$4 x + \frac{π}{3} = 0$

Step 1.2.5

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

$4 x = - \frac{π}{3}$

Step 1.2.6

Divide each term in $4 x = - \frac{π}{3}$ by $4$ and simplify.

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Step 1.2.6.1

Divide each term in $4 x = - \frac{π}{3}$ by $4$ .

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

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.

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

Step 1.2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{3}}{4}$

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.

$x = - \frac{π}{3} \cdot \frac{1}{4}$

Step 1.2.6.3.2

Multiply $- \frac{π}{3} \cdot \frac{1}{4}$ .

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Step 1.2.6.3.2.1

Multiply $\frac{1}{4}$ by $\frac{π}{3}$ .

$x = - \frac{π}{4 \cdot 3}$

Step 1.2.6.3.2.2

Multiply $4$ by $3$ .

$x = - \frac{π}{12}$

Step 1.2.7

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

$4 x + \frac{π}{3} = π - 0$

Step 1.2.8

Solve for $x$ .

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Step 1.2.8.1

Subtract $0$ from $π$ .

$4 x + \frac{π}{3} = π$

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.

$4 x = π - \frac{π}{3}$

Step 1.2.8.2.2

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

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

Step 1.2.8.2.3

Combine $π$ and $\frac{3}{3}$ .

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

Step 1.2.8.2.4

Combine the numerators over the common denominator.

$4 x = \frac{π \cdot 3 - π}{3}$

Step 1.2.8.2.5

Simplify the numerator.

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Step 1.2.8.2.5.1

Move $3$ to the left of $π$ .

$4 x = \frac{3 \cdot π - π}{3}$

Step 1.2.8.2.5.2

Subtract $π$ from $3 π$ .

$4 x = \frac{2 π}{3}$

Step 1.2.8.3

Divide each term in $4 x = \frac{2 π}{3}$ by $4$ and simplify.

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Step 1.2.8.3.1

Divide each term in $4 x = \frac{2 π}{3}$ by $4$ .

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

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.

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

Step 1.2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{2 π}{3}}{4}$

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.

$x = \frac{2 π}{3} \cdot \frac{1}{4}$

Step 1.2.8.3.3.2

Cancel the common factor of $2$ .

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Step 1.2.8.3.3.2.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{3} \cdot \frac{1}{4}$

Step 1.2.8.3.3.2.2

Factor $2$ out of $4$ .

$x = \frac{2 (π)}{3} \cdot \frac{1}{2 (2)}$

Step 1.2.8.3.3.2.3

Cancel the common factor.

$x = \frac{2 π}{3} \cdot \frac{1}{2 \cdot 2}$

Step 1.2.8.3.3.2.4

Rewrite the expression.

$x = \frac{π}{3} \cdot \frac{1}{2}$

Step 1.2.8.3.3.3

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

$x = \frac{π}{3 \cdot 2}$

Step 1.2.8.3.3.4

Multiply $3$ by $2$ .

$x = \frac{π}{6}$

Step 1.2.9

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

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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

Add $\frac{π}{2}$ to every negative angle to get positive angles.

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Step 1.2.10.1

Add $\frac{π}{2}$ to $- \frac{π}{12}$ to find the positive angle.

$- \frac{π}{12} + \frac{π}{2}$

Step 1.2.10.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{π}{2} \cdot \frac{6}{6} - \frac{π}{12}$

Step 1.2.10.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.10.3.1

Multiply $\frac{π}{2}$ by $\frac{6}{6}$ .

$\frac{π \cdot 6}{2 \cdot 6} - \frac{π}{12}$

Step 1.2.10.3.2

Multiply $2$ by $6$ .

$\frac{π \cdot 6}{12} - \frac{π}{12}$

Step 1.2.10.4

Combine the numerators over the common denominator.

$\frac{π \cdot 6 - π}{12}$

Step 1.2.10.5

Simplify the numerator.

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Step 1.2.10.5.1

Move $6$ to the left of $π$ .

$\frac{6 \cdot π - π}{12}$

Step 1.2.10.5.2

Subtract $π$ from $6 π$ .

$\frac{5 π}{12}$

Step 1.2.10.6

List the new angles.

$x = \frac{5 π}{12}$

Step 1.2.11

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

$x = \frac{π}{6} + \frac{π n}{2}, \frac{5 π}{12} + \frac{π n}{2}$ , for any integer $n$

Step 1.2.12

Consolidate the answers.

$x = \frac{π}{6} + \frac{π n}{4}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{6} + \frac{π n}{4}, 0)$ , 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 $y$ .

$y = - 5 \sin (4 (0) + \frac{π}{3})$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 5 \sin (4 (0) + \frac{π}{3})$

Step 2.2.2

Simplify $- 5 \sin (4 (0) + \frac{π}{3})$ .

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Step 2.2.2.1

Multiply $4$ by $0$ .

$y = - 5 \sin (0 + \frac{π}{3})$

Step 2.2.2.2

Add $0$ and $\frac{π}{3}$ .

$y = - 5 \sin (\frac{π}{3})$

Step 2.2.2.3

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

$y = - 5 \frac{\sqrt{3}}{2}$

Step 2.2.2.4

Combine $- 5$ and $\frac{\sqrt{3}}{2}$ .

$y = \frac{- 5 \sqrt{3}}{2}$

Step 2.2.2.5

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3}}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{5 \sqrt{3}}{2})$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{6} + \frac{π n}{4}, 0)$ , for any integer $n$

y-intercept(s): $(0, - \frac{5 \sqrt{3}}{2})$

Step 4