Calculate the period of the function y = cos (5x).
Question
Calculate the period of the function y = cos (5x).
Solution
Para calcular el período de la función , sigue estos pasos:
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Identifica la forma general de la función coseno: La forma general de una función coseno es , donde es una constante que afecta el período de la función.
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Determina el período de la función coseno estándar: La función coseno estándar tiene un período de .
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Ajusta el período según el valor de : El período de la función se calcula dividiendo el período de la función coseno estándar por el valor absoluto de . Es decir, el período es:
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Sustituye el valor de : En la función dada , el valor de es 5. Por lo tanto, sustituimos en la fórmula del período:
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Simplifica la expresión: Calcula el valor del período:
Por lo tanto, el período de la función es .
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Graph the function.g(x) = 5 cos(x)The x y-coordinate plane is given. A curve has a cycle that repeats horizontally every 2𝜋.One cycle starts on the x-axis at x = −𝜋, goes up and right becoming less steep, changes direction at the point (−𝜋⁄2, 5), goes down and right becoming more steep, crosses the x-axis at x = 0, goes down and right becoming less steep, changes direction at the point (𝜋⁄2, −5), goes up and right becoming more steep, and stops on the x-axis at x = 𝜋.The next cycle starts at x = 𝜋. The x y-coordinate plane is given. A curve has a cycle that repeats horizontally every 2𝜋.One cycle starts on the x-axis at x = −𝜋, goes down and right becoming less steep, changes direction at the point (−𝜋⁄2, −5), goes up and right becoming more steep, crosses the x-axis at x = 0, goes up and right becoming less steep, changes direction at the point (𝜋⁄2, 5), goes down and right becoming more steep, and stops on the x-axis at x = 𝜋.The next cycle starts at x = 𝜋. The x y-coordinate plane is given. A curve has a cycle that repeats horizontally every 2𝜋.One cycle starts at the point (−𝜋, 5), goes down and right becoming more steep, crosses the x-axis at x = −𝜋⁄2, goes down and right becoming less steep, crosses the y-axis at y = −5, goes up and right becoming more steep, crosses the x-axis at x = 𝜋⁄2, goes up and right becoming less steep, and stops at the point (𝜋, 5).The next cycle starts at x = 𝜋. The x y-coordinate plane is given. A curve has a cycle that repeats horizontally every 2𝜋.One cycle starts at the point (−𝜋, −5), goes up and right becoming more steep, crosses the x-axis at x = −𝜋⁄2, goes up and right becoming less steep, crosses the y-axis at y = 5, goes down and right becoming more steep, crosses the x-axis at x = 𝜋⁄2, goes down and right becoming less steep, and stops at the point (𝜋, −5).The next cycle starts at x = 𝜋.State the domain and range. (Enter your answers using interval notation.)domain range
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