https://file.aiquickdraw.com/mathbot/file/17320748331726yxpw60g.jpg

from PIL import Image
import pytesseract

# Load the uploaded image to extract text
image_path = '/mnt/data/17320748331726yxpw60g.jpg'
extracted_text = pytesseract.image_to_string(Image.open(image_path))

extracted_text'Sketch the graph of each function.\n\n1) y=4x?\n''Sketch the graph of each function.\n\n1) y=4x?\n'successThe problem is to sketch the graph of the function $ y = 4x^2 $. Let's analyze and describe the function.

### Analysis of $ y = 4x^2 $:
1. **Type of Function**: It is a quadratic function, represented by a parabola.
2. **Vertex**: The vertex is at $ (0, 0) $, since the function is in the form $ y = ax^2 $ where $ a = 4 $.
3. **Direction**: The parabola opens upwards because $ a = 4 > 0 $.
4. **Stretch Factor**: The coefficient $ 4 $ makes the parabola steeper than the standard $ y = x^2 $.

### Key Points for the Graph:
- At $ x = 0 $, $ y = 0 $ (vertex).
- At $ x = 1 $, $ y = 4(1)^2 = 4 $.
- At $ x = -1 $, $ y = 4(-1)^2 = 4 $.
- At $ x = 2 $, $ y = 4(2)^2 = 16 $.
- At $ x = -2 $, $ y = 4(-2)^2 = 16 $.

Let me plot the graph for you.import numpy as np
import matplotlib.pyplot as plt

# Define the function y = 4x^2
x = np.linspace(-3, 3, 500)  # Range of x values
y = 4 * x**2  # Corresponding y values

# Plot the function
plt.figure(figsize=(8, 6))
plt.plot(x, y, label=r"$y = 4x^2$", linewidth=2)
plt.title("Graph of y = 4x²", fontsize=14)
plt.xlabel("x", fontsize=12)
plt.ylabel("y", fontsize=12)
plt.axhline(0, color='black',linewidth=0.8, linestyle='--')
plt.axvline(0, color='black',linewidth=0.8, linestyle='--')
plt.grid(color='gray', linestyle='--', linewidth=0.5)
plt.legend(fontsize=12)
plt.show()
<<ImageDisplayed>>successHere is the graph of the function $ y = 4x^2 $. The parabola opens upwards, and it is steep due to the coefficient $ 4 $.

Would you like further details on this function or another topic? Here are some follow-up questions:

1. How does the coefficient $ 4 $ affect the steepness of the parabola compared to $ y = x^2 $?
2. What is the axis of symmetry for this parabola?
3. How would the graph change if the function was $ y = -4x^2 $?
4. Can you find the derivative of $ y = 4x^2 $ and interpret its meaning?
5. How would the vertex shift if the equation were $ y = 4(x-2)^2 + 3 $?

**Tip:** To sketch a graph by hand, always identify key points (e.g., vertex, axis of symmetry, and a few points on either side of the vertex).

Sketch the graph of the function y = 4x².

Learn how to sketch the graph of the quadratic function y = 4x². This guide explains key concepts such as the vertex, axis of symmetry, and the effect of the coefficient on the graph's shape. Suitable for Grades 9-11.

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