Question

Y = 2 - cosX описать свойства графика

Answer 1

Область определения: x ∈ R
Минимум:  $(2 \pi n, 1)$, n ∈ Z
Максимум:  $( \pi +2 \pi n, 3)$, n ∈ Z
Пересечение с осью y: (0,1)

Answer 2

To describe the properties of the graph of the function $Y = 2 - \cos(X)$, we need to analyze its key features. Let's take a look at some of its characteristics:

1. Periodicity: The graph of $Y = \cos(X)$ has a period of $2\pi$, which means it repeats itself every $2\pi$ units horizontally. However, when we subtract $\cos(X)$ from 2, this affects the graph vertically, but it does not change the periodicity.

2. Amplitude and Range: The cosine function oscillates between -1 and 1. When we subtract $\cos(X)$ from 2, the new function's amplitude is $2 - (-1) = 3$, and its range becomes $[2 - 1, 2 + 1] = [1, 3]$. So, the graph of $Y = 2 - \cos(X)$ will oscillate between 1 and 3.

3. Vertical Shift: The constant term $2$ in the equation causes a vertical shift upward by 2 units. This means that the graph of $Y = 2 - \cos(X)$ will be shifted 2 units up compared to the graph of $Y = \cos(X)$.

4. Symmetry: The cosine function is an even function, meaning $\cos(-X) = \cos(X)$. When we subtract it from 2, the resulting function, $Y = 2 - \cos(X)$, remains even. This implies that the graph is symmetric about the y-axis.

5. Intercepts: To find the y-intercept, let $X = 0$ in the equation: $Y = 2 - \cos(0) = 2 - 1 = 1$. So, the graph passes through the point (0, 1). There are no x-intercepts since $\cos(X)$ never equals 2.

6. Extrema: The cosine function oscillates between -1 and 1, which means the function $2 - \cos(X)$ will have a maximum value of $2 - (-1) = 3$ and a minimum value of $2 - 1 = 1$.

7. Asymptotes: The function $Y = 2 - \cos(X)$ does not have any asymptotes, as it is a continuous function.

8. Periodic Behavior: Since the function $Y = 2 - \cos(X)$ contains $\cos(X)$, it will exhibit the same periodic behavior as the cosine function. It will have repeated peaks and valleys every $2\pi$ units.

9. Graph Shape: Combining all the above properties, the graph of $Y = 2 - \cos(X)$ will be a continuous, smooth, and periodic function that oscillates between 1 and 3, with the central axis of symmetry at $Y = 2$. It will resemble the graph of $Y = \cos(X)$ shifted up by 2 units and vertically stretched by a factor of 3.