Graph y=10cos(pix-pi/2)

$y = 10 \cos (π x - \frac{π}{2})$

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

$a = 10$

$b = π$

$c = \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $10$

Find the period of $10 \cos (π x - \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| π |}$

$π$ is approximately $3.14159265$ which is positive so remove the absolute value

$\frac{2 π}{π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{2 π}{π}$

Divide $2$ by $1$ .

$2$

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

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

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

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

Phase Shift: $\frac{\frac{π}{2}}{π}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $π$ .

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Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{1}{2}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $10$

Period: $2$

Phase Shift: $\frac{1}{2}$ ( $\frac{1}{2}$ to the right)

Vertical Shift: $0$

Select a few points to graph.

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Find the point at $x = \frac{1}{2}$ .

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Replace the variable $x$ with $\frac{1}{2}$ in the expression.

$f (\frac{1}{2}) = 10 \cos (π (\frac{1}{2}) - \frac{π}{2})$

Simplify the result.

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Combine $π$ and $\frac{1}{2}$ .

$f (\frac{1}{2}) = 10 \cos (\frac{π}{2} - \frac{π}{2})$

Combine the numerators over the common denominator.

$f (\frac{1}{2}) = 10 \cos (\frac{π - π}{2})$

Subtract $π$ from $π$ .

$f (\frac{1}{2}) = 10 \cos (\frac{0}{2})$

Divide $0$ by $2$ .

$f (\frac{1}{2}) = 10 \cos (0)$

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

$f (\frac{1}{2}) = 10 \cdot 1$

Multiply $10$ by $1$ .

$f (\frac{1}{2}) = 10$

The final answer is $10$ .

$10$

Find the point at $x = 1$ .

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Replace the variable $x$ with $1$ in the expression.

$f (1) = 10 \cos (π (1) - \frac{π}{2})$

Simplify the result.

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Multiply $π$ by $1$ .

$f (1) = 10 \cos (π - \frac{π}{2})$

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

$f (1) = 10 \cos (π \cdot \frac{2}{2} - \frac{π}{2})$

Combine fractions.

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Combine $π$ and $\frac{2}{2}$ .

$f (1) = 10 \cos (\frac{π \cdot 2}{2} - \frac{π}{2})$

Combine the numerators over the common denominator.

$f (1) = 10 \cos (\frac{π \cdot 2 - π}{2})$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$f (1) = 10 \cos (\frac{2 \cdot π - π}{2})$

Subtract $π$ from $2 π$ .

$f (1) = 10 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (1) = 10 \cdot 0$

Multiply $10$ by $0$ .

$f (1) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{3}{2}$ .

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Replace the variable $x$ with $\frac{3}{2}$ in the expression.

$f (\frac{3}{2}) = 10 \cos (π (\frac{3}{2}) - \frac{π}{2})$

Simplify the result.

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Simplify each term.

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Combine $π$ and $\frac{3}{2}$ .

$f (\frac{3}{2}) = 10 \cos (\frac{π \cdot 3}{2} - \frac{π}{2})$

Move $3$ to the left of $π$ .

$f (\frac{3}{2}) = 10 \cos (\frac{3 π}{2} - \frac{π}{2})$

Simplify terms.

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Combine the numerators over the common denominator.

$f (\frac{3}{2}) = 10 \cos (\frac{3 π - π}{2})$

Subtract $π$ from $3 π$ .

$f (\frac{3}{2}) = 10 \cos (\frac{2 π}{2})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (\frac{3}{2}) = 10 \cos (\frac{2 π}{2})$

Divide $π$ by $1$ .

$f (\frac{3}{2}) = 10 \cos (π)$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (\frac{3}{2}) = 10 (- \cos (0))$

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

$f (\frac{3}{2}) = 10 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$f (\frac{3}{2}) = 10 \cdot - 1$

Multiply $10$ by $- 1$ .

$f (\frac{3}{2}) = - 10$

The final answer is $- 10$ .

$- 10$

Find the point at $x = 2$ .

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Replace the variable $x$ with $2$ in the expression.

$f (2) = 10 \cos (π (2) - \frac{π}{2})$

Simplify the result.

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Move $2$ to the left of $π$ .

$f (2) = 10 \cos (2 π - \frac{π}{2})$

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

$f (2) = 10 \cos (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Combine fractions.

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Combine $2 π$ and $\frac{2}{2}$ .

$f (2) = 10 \cos (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Combine the numerators over the common denominator.

$f (2) = 10 \cos (\frac{2 π \cdot 2 - π}{2})$

Simplify the numerator.

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Multiply $2$ by $2$ .

$f (2) = 10 \cos (\frac{4 π - π}{2})$

Subtract $π$ from $4 π$ .

$f (2) = 10 \cos (\frac{3 π}{2})$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (2) = 10 \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (2) = 10 \cdot 0$

Multiply $10$ by $0$ .

$f (2) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{5}{2}$ .

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Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = 10 \cos (π (\frac{5}{2}) - \frac{π}{2})$

Simplify the result.

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Simplify each term.

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Combine $π$ and $\frac{5}{2}$ .

$f (\frac{5}{2}) = 10 \cos (\frac{π \cdot 5}{2} - \frac{π}{2})$

Move $5$ to the left of $π$ .

$f (\frac{5}{2}) = 10 \cos (\frac{5 π}{2} - \frac{π}{2})$

Simplify terms.

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Combine the numerators over the common denominator.

$f (\frac{5}{2}) = 10 \cos (\frac{5 π - π}{2})$

Subtract $π$ from $5 π$ .

$f (\frac{5}{2}) = 10 \cos (\frac{4 π}{2})$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4 π$ .

$f (\frac{5}{2}) = 10 \cos (\frac{2 (2 π)}{2})$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (\frac{5}{2}) = 10 \cos (\frac{2 (2 π)}{2 (1)})$

Cancel the common factor.

$f (\frac{5}{2}) = 10 \cos (\frac{2 (2 π)}{2 \cdot 1})$

Rewrite the expression.

$f (\frac{5}{2}) = 10 \cos (\frac{2 π}{1})$

Divide $2 π$ by $1$ .

$f (\frac{5}{2}) = 10 \cos (2 π)$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{5}{2}) = 10 \cos (0)$

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

$f (\frac{5}{2}) = 10 \cdot 1$

Multiply $10$ by $1$ .

$f (\frac{5}{2}) = 10$

The final answer is $10$ .

$10$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{1}{2} & 10 \\ 1 & 0 \\ \frac{3}{2} & - 10 \\ 2 & 0 \\ \frac{5}{2} & 10 \end{array}$

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

Amplitude: $10$

Period: $2$

Phase Shift: $\frac{1}{2}$ ( $\frac{1}{2}$ to the right)

Vertical Shift: $0$

$\begin{array}{c} x & f (x) \\ \frac{1}{2} & 10 \\ 1 & 0 \\ \frac{3}{2} & - 10 \\ 2 & 0 \\ \frac{5}{2} & 10 \end{array}$

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