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).