Pre-Algebra Examples

$3 \sin (x - 3.14) - 1$

Step 1

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

$a = 3$

$b = 1$

$c = 3.14$

$d = - 1$

Step 2

Find the amplitude $| a |$ .

Amplitude: $3$

Step 3

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

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Step 3.1

Find the period of $3 \sin (x - 3.14)$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.1.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2

Find the period of $- 1$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.3

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

$2 π$

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: $\frac{3.14}{1}$

Step 4.3

Divide $3.14$ by $1$ .

Phase Shift: $3.14$

Step 5

List the properties of the trigonometric function.

Amplitude: $3$

Period: $2 π$

Phase Shift: $3.14$ ( $3.14$ to the right)

Vertical Shift: $- 1$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 3.14$ .

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Step 6.1.1

Replace the variable $x$ with $3.14$ in the expression.

$f (3.14) = 3 \sin ((3.14) - 3.14) - 1$

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

Subtract $3.14$ from $3.14$ .

$f (3.14) = 3 \sin (0) - 1$

Step 6.1.2.1.2

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

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Step 6.1.2.1.2.1

Rewrite $0$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$f (3.14) = 3 \sin (\frac{0}{2}) - 1$

Step 6.1.2.1.2.2

Apply the sine half-angle identity.

$f (3.14) = 3 (\pm \sqrt{\frac{1 - \cos (0)}{2}}) - 1$

Step 6.1.2.1.2.3

Change the $\pm$ to $+$ because sine is positive in the first quadrant.

$f (3.14) = 3 \sqrt{\frac{1 - \cos (0)}{2}} - 1$

Step 6.1.2.1.2.4

Simplify $\sqrt{\frac{1 - \cos (0)}{2}}$ .

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Step 6.1.2.1.2.4.1

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

$f (3.14) = 3 \sqrt{\frac{1 - 1 \cdot 1}{2}} - 1$

Step 6.1.2.1.2.4.2

Multiply $- 1$ by $1$ .

$f (3.14) = 3 \sqrt{\frac{1 - 1}{2}} - 1$

Step 6.1.2.1.2.4.3

Subtract $1$ from $1$ .

$f (3.14) = 3 \sqrt{\frac{0}{2}} - 1$

Step 6.1.2.1.2.4.4

Divide $0$ by $2$ .

$f (3.14) = 3 \sqrt{0} - 1$

Step 6.1.2.1.2.4.5

Rewrite $0$ as $0^{2}$ .

$f (3.14) = 3 \sqrt{0^{2}} - 1$

Step 6.1.2.1.2.4.6

Pull terms out from under the radical, assuming positive real numbers.

$f (3.14) = 3 \cdot 0 - 1$

Step 6.1.2.1.3

Multiply $3$ by $0$ .

$f (3.14) = 0 - 1$

Step 6.1.2.2

Subtract $1$ from $0$ .

$f (3.14) = - 1$

Step 6.1.2.3

The final answer is $- 1$ .

$- 1$

Step 6.2

Find the point at $x = 4.71079632$ .

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Step 6.2.1

Replace the variable $x$ with $4.71079632$ in the expression.

$f (4.71079632) = 3 \sin ((4.71079632) - 3.14) - 1$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Subtract $3.14$ from $4.71079632$ .

$f (4.71079632) = 3 \sin (1.57079632) - 1$

Step 6.2.2.2

The final answer is $3 \sin (1.57079632) - 1$ .

$3 \sin (1.57079632) - 1$

Step 6.2.3

Convert $3 \sin (1.57079632) - 1$ to a decimal.

$2$

Step 6.3

Find the point at $x = 6.28159265$ .

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Step 6.3.1

Replace the variable $x$ with $6.28159265$ in the expression.

$f (6.28159265) = 3 \sin ((6.28159265) - 3.14) - 1$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Subtract $3.14$ from $6.28159265$ .

$f (6.28159265) = 3 \sin (3.14159265) - 1$

Step 6.3.2.2

The final answer is $3 \sin (3.14159265) - 1$ .

$3 \sin (3.14159265) - 1$

Step 6.3.3

Convert $3 \sin (3.14159265) - 1$ to a decimal.

$- 1$

Step 6.4

Find the point at $x = 7.85238898$ .

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Step 6.4.1

Replace the variable $x$ with $7.85238898$ in the expression.

$f (7.85238898) = 3 \sin ((7.85238898) - 3.14) - 1$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Subtract $3.14$ from $7.85238898$ .

$f (7.85238898) = 3 \sin (4.71238898) - 1$

Step 6.4.2.2

The final answer is $3 \sin (4.71238898) - 1$ .

$3 \sin (4.71238898) - 1$

Step 6.4.3

Convert $3 \sin (4.71238898) - 1$ to a decimal.

$- 4$

Step 6.5

Find the point at $x = 9.4231853$ .

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Step 6.5.1

Replace the variable $x$ with $9.4231853$ in the expression.

$f (9.4231853) = 3 \sin ((9.4231853) - 3.14) - 1$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Subtract $3.14$ from $9.4231853$ .

$f (9.4231853) = 3 \sin (6.2831853) - 1$

Step 6.5.2.2

The final answer is $3 \sin (6.2831853) - 1$ .

$3 \sin (6.2831853) - 1$

Step 6.5.3

Convert $3 \sin (6.2831853) - 1$ to a decimal.

$- 1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 3.14 & - 1 \\ 4.711 & 2 \\ 6.282 & - 1 \\ 7.852 & - 4 \\ 9.423 & - 1 \end{array}$

Step 7

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

Amplitude: $3$

Period: $2 π$

Phase Shift: $3.14$ ( $3.14$ to the right)

Vertical Shift: $- 1$

$\begin{array}{c} x & f (x) \\ 3.14 & - 1 \\ 4.711 & 2 \\ 6.282 & - 1 \\ 7.852 & - 4 \\ 9.423 & - 1 \end{array}$

Step 8