Find the X and Y Intercepts f(x)=4cos(2x-pi)

$f (x) = 4 \cos (2 x - π)$

To find the x-intercept, substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 \cos (2 x - π)$

Solve the equation.

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Rewrite the equation as $4 \cos (2 x - π) = 0$ .

$4 \cos (2 x - π) = 0$

Divide each term by $4$ and simplify.

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Divide each term in $4 \cos (2 x - π) = 0$ by $4$ .

$\frac{4 \cos (2 x - π)}{4} = \frac{0}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 \cos (2 x - π)}{4} = \frac{0}{4}$

Divide $\cos (2 x - π)$ by $1$ .

$\cos (2 x - π) = \frac{0}{4}$

Divide $0$ by $4$ .

$\cos (2 x - π) = 0$

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

$2 x - π = \arccos (0)$

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

$2 x - π = \frac{π}{2}$

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

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Add $π$ to both sides of the equation.

$2 x = \frac{π}{2} + π$

Simplify the right side of the equation.

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = \frac{π}{2} + \frac{π}{1} \cdot \frac{2}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$2 x = \frac{π}{2} + \frac{π \cdot 2}{1 \cdot 2}$

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

$2 x = \frac{π + π \cdot 2}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$2 x = \frac{π + 2 \cdot π}{2}$

Add $π$ and $2 π$ .

$2 x = \frac{3 π}{2}$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{3 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{3 π}{2} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{3 π}{2} \cdot \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{3 π}{2} \cdot \frac{1}{2}$

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{3 π}{2}$ and $\frac{1}{2}$ .

$x = \frac{3 π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x - π = 2 π - \frac{π}{2}$

Simplify the expression to find the second solution.

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Simplify $2 π - \frac{π}{2}$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x - π = \frac{2 π}{1} \cdot \frac{2}{2} - \frac{π}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$2 x - π = \frac{2 π \cdot 2}{1 \cdot 2} - \frac{π}{2}$

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

$2 x - π = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$2 x - π = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$2 x - π = \frac{3 π}{2}$

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

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Add $π$ to both sides of the equation.

$2 x = \frac{3 π}{2} + π$

Simplify the right side of the equation.

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = \frac{3 π}{2} + \frac{π}{1} \cdot \frac{2}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$2 x = \frac{3 π}{2} + \frac{π \cdot 2}{1 \cdot 2}$

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

$2 x = \frac{3 π + π \cdot 2}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$2 x = \frac{3 π + 2 \cdot π}{2}$

Add $3 π$ and $2 π$ .

$2 x = \frac{5 π}{2}$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{5 π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{5 π}{2} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{5 π}{2} \cdot \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{5 π}{2} \cdot \frac{1}{2}$

Multiply $\frac{5 π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{5 π}{2}$ and $\frac{1}{2}$ .

$x = \frac{5 π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{5 π}{4}$

Find the period.

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 2 |}$

Solve the equation.

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

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

$x = \frac{3 π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

To find the y-intercept, substitute in $0$ for $x$ and solve for $y$ .

$y = 4 \cos (2 (0) - π)$

Simplify $4 \cos (2 (0) - π)$ .

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Multiply $2$ by $0$ .

$y = 4 \cos (0 - π)$

Subtract $π$ from $0$ .

$y = 4 \cos (- π)$

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.

$y = 4 (- \cos (0))$

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

$y = 4 (- 1 \cdot 1)$

Multiply $- 1$ by $1$ .

$y = 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$y = - 4$

These are the $x$ and $y$ intercepts of the equation $y = 4 \cos (2 x - π)$ .

x-intercept: $(\frac{π}{4} + \frac{π n}{2}, 0)$ , for any integer $n$

y-intercept: $(0, - 4)$

Find the X and Y Intercepts f(x)=4cos(2x-pi)

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