Question

Постройте график функций y=(x-4)в квадрате+1.

Answer 1

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Answer 2

Plot of the Function y=(x-4)^2+1

To plot the function \(y=(x-4)^2+1\), we can use the information provided in the search results to understand the behavior of the function and identify key points.

The function \(y=(x-4)^2+1\) represents a parabola with its vertex at the point (4, 1). The parabola opens upwards because the coefficient of \(x^2\) is positive.

Key Points:

- Vertex: The vertex of the parabola is at the point (4, 1). - x-intercept: To find the x-intercept, we can set y=0 and solve for x. From the search results, we find that the x-intercept is at \(x = \frac{5}{2}\).

Plotting the Function:

We can use this information to plot the function on a graph, with the vertex at (4, 1) and the x-intercept at \(\frac{5}{2}\).

import matplotlib.pyplot as plt
import numpy as np

# Define the function
def f(x):
    return (x-4)**2 + 1

# Generate x values
x = np.linspace(-2, 10, 100)
# Generate y values
y = f(x)

# Plot the function
plt.figure(figsize=(8, 6))
plt.plot(x, y, label='$y=(x-4)^2+1$')
plt.scatter(4, 1, color='red', label='Vertex (4, 1)')
plt.scatter(2.5, 0, color='green', label='X-Intercept (2.5, 0)')
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of $y=(x-4)^2+1$')
plt.legend()
plt.grid(True)
plt.show()

The graph of the function \(y=(x-4)^2+1\) will show a parabola opening upwards with the vertex at (4, 1) and an x-intercept at \(\frac{5}{2}\).