Función Coseno. Sus características. Dominio e imagen, ceros y período

Summary

This video explains the cosine function, covering its graph, domain, and range. The cosine function is periodic, with a period of 2π and a range from -1 to 1. It is symmetric about the y-axis, making it an even function. Key concepts include the zeros of the cosine function, where it intersects the x-axis, and the intervals where the function is increasing, decreasing, or positive. The lesson also discusses the amplitude and how scaling the function affects its range, focusing on a detailed understanding of the cosine function's behavior.

Takeaways

• 😀 The cosine function is periodic, similar to the sine function, and its graph represents a wave-like curve.
• 😀 The domain of the cosine function is all real numbers, from negative infinity to positive infinity.
• 😀 The range (image) of the cosine function is between -1 and 1, meaning it never exceeds these values.
• 😀 The period of the cosine function is 2π, meaning the values repeat every 2π units.
• 😀 The cosine function intersects the x-axis (has zeros) at multiples of π/2, such as π/2, 3π/2, and so on.
• 😀 Cosine is an even function, meaning cos(-x) = cos(x) for all x in its domain.
• 😀 The symmetry of the cosine function about the y-axis demonstrates its evenness, which can be seen in the graph.
• 😀 The cosine function decreases from 1 to -1 as x moves from 0 to π, and increases from -1 to 1 from π to 2π.
• 😀 The cosine function is positive between 0 and π/2, and between 3π/2 and 2π. It is negative between π/2 and 3π/2.
• 😀 If the cosine function is scaled by a factor, such as 5 cos(x), the amplitude changes, but the periodicity and shape remain the same.

Q & A

What is the domain of the cosine function?

The domain of the cosine function is all real numbers, meaning it extends from negative infinity to positive infinity.

What is the range (image) of the cosine function?

The range of the cosine function is between -1 and 1, meaning its values are confined within this interval for all x values.

What is the period of the cosine function?

The period of the cosine function is 2π, meaning it repeats its values every 2π units along the x-axis.

Where are the zeros of the cosine function located?

The zeros of the cosine function occur at odd multiples of π/2, such as π/2, 3π/2, 5π/2, and so on.

What does the symmetry of the cosine function imply?

The cosine function is an even function, meaning it is symmetric with respect to the y-axis. This means that cos(x) = cos(-x) for all x.

How does the cosine function behave between 0 and π/2?

Between 0 and π/2, the cosine function is positive and decreases from 1 to 0.

How does the cosine function behave between π/2 and 3π/2?

Between π/2 and 3π/2, the cosine function is negative and increases from -1 to 0.

What is the significance of the amplitude of the cosine function?

The amplitude of the cosine function is the maximum absolute value of the function, which is 1 for the standard cosine function. It can be altered if the function is scaled by a constant factor.

What happens if the cosine function is scaled by a factor, like 5?

If the cosine function is scaled by a factor of 5, the amplitude will change from 1 to 5, and the range will extend from -5 to 5 instead of -1 to 1.

What is meant by the term 'periodicity' in the context of the cosine function?

Periodicity refers to the property of the cosine function repeating its values in regular intervals. In this case, the cosine function repeats every 2π units along the x-axis.