Calculus Examples

$- \cos (x \cdot 2.4) + 2$

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.

$a = - 1$

$b = 2.4$

$c = 0$

$d = 2$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

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Step 3.1

Find the period of $- \cos (2.4 x)$ .

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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 $2.4$ in the formula for period.

$\frac{2 π}{| 2.4 |}$

Step 3.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2.4$ is $2.4$ .

$\frac{2 π}{2.4}$

Step 3.1.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{2.4}$

Step 3.1.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{2.4}$

Step 3.1.6

Divide $6.2831853$ by $2.4$ .

$2.61799387$

Step 3.2

Find the period of $2$ .

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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 $2.4$ in the formula for period.

$\frac{2 π}{| 2.4 |}$

Step 3.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2.4$ is $2.4$ .

$\frac{2 π}{2.4}$

Step 3.2.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{2.4}$

Step 3.2.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{2.4}$

Step 3.2.6

Divide $6.2831853$ by $2.4$ .

$2.61799387$

Step 3.3

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

$2.61799387$

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: $\frac{0}{2.4}$

Step 4.3

Divide $0$ by $2.4$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2.61799387$

Phase Shift: None

Vertical Shift: $2$

Step 6

Select a few points to graph.

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Step 6.1

Find the point at $x = 0$ .

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Step 6.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \cos (2.4 (0)) + 2$

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

Multiply $2.4$ by $0$ .

$f (0) = - \cos (0) + 2$

Step 6.1.2.1.2

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

$f (0) = - 1 \cdot 1 + 2$

Step 6.1.2.1.3

Multiply $- 1$ by $1$ .

$f (0) = - 1 + 2$

Step 6.1.2.2

Add $- 1$ and $2$ .

$f (0) = 1$

Step 6.1.2.3

The final answer is $1$ .

$1$

Step 6.2

Find the point at $x = 0.65449846$ .

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Step 6.2.1

Replace the variable $x$ with $0.65449846$ in the expression.

$f (0.65449846) = - \cos (2.4 (0.65449846)) + 2$

Step 6.2.2

Simplify the result.

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Step 6.2.2.1

Multiply $2.4$ by $0.65449846$ .

$f (0.65449846) = - \cos (1.57079632) + 2$

Step 6.2.2.2

The final answer is $- \cos (1.57079632) + 2$ .

$- \cos (1.57079632) + 2$

Step 6.2.3

Convert $- \cos (1.57079632) + 2$ to a decimal.

$2$

Step 6.3

Find the point at $x = 1.30899693$ .

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Step 6.3.1

Replace the variable $x$ with $1.30899693$ in the expression.

$f (1.30899693) = - \cos (2.4 (1.30899693)) + 2$

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Multiply $2.4$ by $1.30899693$ .

$f (1.30899693) = - \cos (3.14159265) + 2$

Step 6.3.2.2

The final answer is $- \cos (3.14159265) + 2$ .

$- \cos (3.14159265) + 2$

Step 6.3.3

Convert $- \cos (3.14159265) + 2$ to a decimal.

$3$

Step 6.4

Find the point at $x = 1.9634954$ .

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Step 6.4.1

Replace the variable $x$ with $1.9634954$ in the expression.

$f (1.9634954) = - \cos (2.4 (1.9634954)) + 2$

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Multiply $2.4$ by $1.9634954$ .

$f (1.9634954) = - \cos (4.71238898) + 2$

Step 6.4.2.2

The final answer is $- \cos (4.71238898) + 2$ .

$- \cos (4.71238898) + 2$

Step 6.4.3

Convert $- \cos (4.71238898) + 2$ to a decimal.

$2$

Step 6.5

Find the point at $x = 2.61799387$ .

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Step 6.5.1

Replace the variable $x$ with $2.61799387$ in the expression.

$f (2.61799387) = - \cos (2.4 (2.61799387)) + 2$

Step 6.5.2

Simplify the result.

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Step 6.5.2.1

Multiply $2.4$ by $2.61799387$ .

$f (2.61799387) = - \cos (6.2831853) + 2$

Step 6.5.2.2

The final answer is $- \cos (6.2831853) + 2$ .

$- \cos (6.2831853) + 2$

Step 6.5.3

Convert $- \cos (6.2831853) + 2$ to a decimal.

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ 0.654 & 2 \\ 1.309 & 3 \\ 1.963 & 2 \\ 2.618 & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $2.61799387$

Phase Shift: None

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ 0.654 & 2 \\ 1.309 & 3 \\ 1.963 & 2 \\ 2.618 & 1 \end{array}$

Step 8