The graph of $y = f(3x)$, where $f(x) = x^2$, is a horizontally compressed version of the parent function $y = x^2$. Specifically, the function $y = f(3x)$ compresses the graph by a factor of $1/3$, meaning that for each point on the graph, the value of $x$ must be three times as large to produce the same value of $y$ as in the parent function. This results in the parabola becoming narrower.

Key observations:

1. Parent function: $y = x^2$ is a standard parabola with vertex at (0, 0).
2. Compressed graph: The function $y = (3x)^2 = 9x^2$ compresses the graph horizontally. For instance, while the parent function takes the value $1$ at $x = 1$, the compressed function takes the value $1$ at $x = 1/3$.

Let me generate the graph for you to visually represent this transformation. numpy as np import matplotlib.pyplot as plt

Define the functions

x = np.linspace(-3, 3, 400) y_parent = x**2 y_compressed = (3*x)**2

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x, y_parent, label=r'$y = x^2$', color='blue') plt.plot(x, y_compressed, label=r'$y = (3x)^2$', color='red', linestyle='dashed')

Add title and labels

plt.title('Graph of $y = x^2$ and $y = (3x)^2$ (Horizontally Compressed)', fontsize=14) plt.xlabel('x', fontsize=12) plt.ylabel('y', fontsize=12)

Add legend

plt.legend()

Display grid

plt.grid(True)

Show the plot

plt.show()Here is the graph that shows the transformation of the parent function $y = x^2$ to $y = (3x)^2$, which is a horizontally compressed version of the parent function by a factor of $\frac{1}{3}$. The blue curve represents the parent function, and the red dashed curve represents the compressed version.

Would you like further details on the transformation, or do you have any questions?

5 Related Questions:

1. How do vertical transformations affect the graph of $y = x^2$?
2. What happens to the graph if we replace $x$ with $x/3$ in the function?
3. How do horizontal compressions and expansions differ?
4. What are some real-world examples of horizontal compression in graphs?
5. How can you determine the factor of horizontal compression just by looking at the equation?

Tip: To determine the compression factor, look at the coefficient of $x$ inside the function. For $f(3x)$, the compression factor is $\frac{1}{3}$.