Precalculus Examples

$f (x) = 3 \sin (2 x)$

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 $x$ .

$0 = 3 \sin (2 x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 \sin (2 x) = 0$ .

$3 \sin (2 x) = 0$

Step 1.2.2

Divide each term in $3 \sin (2 x) = 0$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 \sin (2 x) = 0$ by $3$ .

$\frac{3 \sin (2 x)}{3} = \frac{0}{3}$

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.

$\frac{3 \sin (2 x)}{3} = \frac{0}{3}$

Step 1.2.2.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{0}{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $3$ .

$\sin (2 x) = 0$

Step 1.2.3

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

$2 x = \arcsin (0)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

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

$2 x = 0$

Step 1.2.5

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 1.2.5.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $0$ by $2$ .

$x = 0$

Step 1.2.6

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.

$2 x = π - 0$

Step 1.2.7

Solve for $x$ .

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Step 1.2.7.1

Simplify.

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Step 1.2.7.1.1

Multiply $- 1$ by $0$ .

$2 x = π + 0$

Step 1.2.7.1.2

Add $π$ and $0$ .

$2 x = π$

Step 1.2.7.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 1.2.7.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 1.2.7.2.2

Simplify the left side.

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Step 1.2.7.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 1.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 1.2.8

Find the period of $\sin (2 x)$ .

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Step 1.2.8.1

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

$\frac{2 π}{| b |}$

Step 1.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 1.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 1.2.8.4

Cancel the common factor of $2$ .

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Step 1.2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 1.2.8.4.2

Divide $π$ by $1$ .

$π$

Step 1.2.9

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

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

Step 1.2.10

Consolidate the answers.

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π n}{2}, 0)$ , 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 $y$ .

$y = 3 \sin (2 (0))$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 3 \sin (2 (0))$

Step 2.2.2

Simplify $3 \sin (2 (0))$ .

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Step 2.2.2.1

Multiply $2$ by $0$ .

$y = 3 \sin (0)$

Step 2.2.2.2

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

$y = 3 \cdot 0$

Step 2.2.2.3

Multiply $3$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(\frac{π n}{2}, 0)$ , for any integer $n$

y-intercept(s): $(0, 0)$

Step 4