The function $f(x) = 3\sqrt{x} + 4$ involves a square root transformation combined with scaling and vertical shifting. To plot four points on this function, we select values for $x$, compute $f(x)$, and then generate the corresponding $(x, f(x))$ points.

Calculations

For $f(x) = 3\sqrt{x} + 4$:

1. Choose $x$ values that allow the square root $\sqrt{x}$ to be easily calculated (e.g., $x = 0, 1, 4, 9$).

Compute $f(x)$:

• For $x = 0$: $f(0) = 3\sqrt{0} + 4 = 4$
• For $x = 1$: $f(1) = 3\sqrt{1} + 4 = 7$
• For $x = 4$: $f(4) = 3\sqrt{4} + 4 = 10$
• For $x = 9$: $f(9) = 3\sqrt{9} + 4 = 13$

Points

The points are:

1. $(0, 4)$
2. $(1, 7)$
3. $(4, 10)$
4. $(9, 13)$

Let me now create a plot for these points. matplotlib.pyplot as plt import numpy as np

Define the function

def f(x): return 3 * np.sqrt(x) + 4

Generate x values and corresponding y values for the points

x_points = [0, 1, 4, 9] y_points = [f(x) for x in x_points]

Generate smooth curve for the function

x_curve = np.linspace(0, 10, 500) # x values for a smooth plot y_curve = f(x_curve)

Plot the function and points

plt.figure(figsize=(8, 6)) plt.plot(x_curve, y_curve, label=r"$f(x) = 3\sqrt{x} + 4$", color='blue') # Function curve plt.scatter(x_points, y_points, color='red', label="Points") # Points

Adding labels and grid

plt.title("Plot of $f(x) = 3\sqrt{x} + 4$", fontsize=14) plt.xlabel("x", fontsize=12) plt.ylabel("f(x)", fontsize=12) plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--') plt.grid(alpha=0.3) plt.legend(fontsize=12) plt.show()Here is the plot for the function $f(x) = 3\sqrt{x} + 4$, along with the four points $(0, 4)$, $(1, 7)$, $(4, 10)$, and $(9, 13)$. The red dots represent the selected points, and the blue curve represents the function.

Would you like me to calculate additional points, explain the graph further, or assist with related questions?

Relative Questions:

1. How does the parameter $3$ affect the shape of the square root function?
2. Why is the domain of $f(x) = 3\sqrt{x} + 4$ restricted to non-negative $x$?
3. How would the graph change if the constant $+4$ was altered to another value?
4. What happens to the graph if the coefficient of $\sqrt{x}$ is negative?
5. How does the growth rate of $\sqrt{x}$ compare to a linear function?

Tip: To better understand the behavior of functions, try plotting multiple variations with different coefficients and constants!