Pre-Algebra Examples

$y = - 2 \sec (\frac{π}{2} x + π) + 2$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \sec (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = s e c (x)$ , $(- \frac{π}{2}, \frac{3 π}{2})$ , to find the vertical asymptotes for $y = - 2 \sec (\frac{π x}{2} + π) + 2$ . Set the inside of the secant function, $b x + c$ , for $y = a \sec (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = - 2 \sec (\frac{π x}{2} + π) + 2$ .

$\frac{π x}{2} + π = - \frac{π}{2}$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

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Step 1.2.1.1

Subtract $π$ from both sides of the equation.

$\frac{π x}{2} = - \frac{π}{2} - π$

Step 1.2.1.2

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

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

Step 1.2.1.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 1.2.1.4

Combine the numerators over the common denominator.

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

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $2$ by $- 1$ .

$\frac{π x}{2} = \frac{- π - 2 π}{2}$

Step 1.2.1.5.2

Subtract $2 π$ from $- π$ .

$\frac{π x}{2} = \frac{- 3 π}{2}$

Step 1.2.1.6

Move the negative in front of the fraction.

$\frac{π x}{2} = - \frac{3 π}{2}$

Step 1.2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$π x = - 3 π$

Step 1.2.3

Divide each term in $π x = - 3 π$ by $π$ and simplify.

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Step 1.2.3.1

Divide each term in $π x = - 3 π$ by $π$ .

$\frac{π x}{π} = \frac{- 3 π}{π}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $π$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{- 3 π}{π}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3 π}{π}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $π$ .

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Step 1.2.3.3.1.1

Cancel the common factor.

$x = \frac{- 3 π}{π}$

Step 1.2.3.3.1.2

Divide $- 3$ by $1$ .

$x = - 3$

Step 1.3

Set the inside of the secant function $\frac{π x}{2} + π$ equal to $\frac{3 π}{2}$ .

$\frac{π x}{2} + π = \frac{3 π}{2}$

Step 1.4

Solve for $x$ .

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Step 1.4.1

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

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Step 1.4.1.1

Subtract $π$ from both sides of the equation.

$\frac{π x}{2} = \frac{3 π}{2} - π$

Step 1.4.1.2

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

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

Step 1.4.1.3

Combine $- π$ and $\frac{2}{2}$ .

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

Step 1.4.1.4

Combine the numerators over the common denominator.

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

Step 1.4.1.5

Simplify the numerator.

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Step 1.4.1.5.1

Multiply $2$ by $- 1$ .

$\frac{π x}{2} = \frac{3 π - 2 π}{2}$

Step 1.4.1.5.2

Subtract $2 π$ from $3 π$ .

$\frac{π x}{2} = \frac{π}{2}$

Step 1.4.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$π x = π$

Step 1.4.3

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

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Step 1.4.3.1

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

$\frac{π x}{π} = \frac{π}{π}$

Step 1.4.3.2

Simplify the left side.

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Step 1.4.3.2.1

Cancel the common factor of $π$ .

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Step 1.4.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{π}{π}$

Step 1.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{π}$

Step 1.4.3.3

Simplify the right side.

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Step 1.4.3.3.1

Cancel the common factor of $π$ .

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Step 1.4.3.3.1.1

Cancel the common factor.

$x = \frac{π}{π}$

Step 1.4.3.3.1.2

Rewrite the expression.

$x = 1$

Step 1.5

The basic period for $y = - 2 \sec (\frac{π x}{2} + π) + 2$ will occur at $(- 3, 1)$ , where $- 3$ and $1$ are vertical asymptotes.

$(- 3, 1)$

Step 1.6

Find the period $\frac{2 π}{| b |}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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Step 1.6.1

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 1.6.3

Cancel the common factor of $π$ .

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Step 1.6.3.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 1.6.3.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 1.6.3.3

Rewrite the expression.

$2 \cdot 2$

Step 1.6.4

Multiply $2$ by $2$ .

$4$

Step 1.7

The vertical asymptotes for $y = - 2 \sec (\frac{π x}{2} + π) + 2$ occur at $- 3$ , $1$ , and every $x = - 3 + 2 n$ , where $n$ is an integer. This is half of the period.

$x = - 3 + 2 n$

Step 1.8

Secant only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - 3 + 2 n$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - 3 + 2 n$ where $n$ is an integer

Step 2

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

$a = - 2$

$b = \frac{π}{2}$

$c = - π$

$d = 2$

Step 3

Since the graph of the function $s e c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

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

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Step 4.1

Find the period of $- 2 \sec (\frac{π x}{2} + π)$ .

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Step 4.1.1

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

$\frac{2 π}{| b |}$

Step 4.1.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 4.1.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 4.1.5

Cancel the common factor of $π$ .

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Step 4.1.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 4.1.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 4.1.5.3

Rewrite the expression.

$2 \cdot 2$

Step 4.1.6

Multiply $2$ by $2$ .

$4$

Step 4.2

Find the period of $2$ .

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Step 4.2.1

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

$\frac{2 π}{| b |}$

Step 4.2.2

Replace $b$ with $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 4.2.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 4.2.5

Cancel the common factor of $π$ .

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Step 4.2.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 4.2.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 4.2.5.3

Rewrite the expression.

$2 \cdot 2$

Step 4.2.6

Multiply $2$ by $2$ .

$4$

Step 4.3

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

$4$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- π}{\frac{π}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- π \frac{2}{π}$

Step 5.4

Cancel the common factor of $π$ .

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Step 5.4.1

Factor $π$ out of $- π$ .

Phase Shift: $π \cdot (- 1 \frac{2}{π})$

Step 5.4.2

Cancel the common factor.

Phase Shift: $π \cdot (- 1 \frac{2}{π})$

Step 5.4.3

Rewrite the expression.

Phase Shift: $- 1 \cdot 2$

Step 5.5

Multiply $- 1$ by $2$ .

Phase Shift: $- 2$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $4$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: $2$

Step 7

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

Vertical Asymptotes: $x = - 3 + 2 n$ where $n$ is an integer

Amplitude: None

Period: $4$

Phase Shift: $- 2$ ( $2$ to the left)

Vertical Shift: $2$

Step 8