The function you provided is:

$f(x) = (x - 3)^2 + 1$

This is a quadratic function, which represents a parabola. To graph it, we can identify some key features:

1. Vertex: The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In this case, $a = 1$, $h = 3$, and $k = 1$, so the vertex is at $(3, 1)$.

2. Axis of symmetry: This is the vertical line passing through the vertex. In this case, the axis of symmetry is $x = 3$.

3. Shape: Since the coefficient $a$ is positive, the parabola opens upwards.

4. Y-intercept: To find the y-intercept, set $x = 0$:

   $f(0) = (0 - 3)^2 + 1 = 9 + 1 = 10$

   So the y-intercept is at $(0, 10)$.

Now, I will graph this function.

Please hold on a moment! numpy as np import matplotlib.pyplot as plt

Define the function f(x) = (x - 3)^2 + 1

def f(x): return (x - 3)**2 + 1

Generate x values from -2 to 8

x = np.linspace(-2, 8, 400) y = f(x)

Plot the graph

plt.figure(figsize=(6,6)) plt.plot(x, y, label=r'$f(x) = (x - 3)^2 + 1$', color='blue') plt.scatter(3, 1, color='red', zorder=5, label="Vertex (3, 1)") plt.scatter(0, 10, color='green', zorder=5, label="Y-intercept (0, 10)") plt.axvline(x=3, color='gray', linestyle='--', label="Axis of Symmetry x=3")

Labels and title

plt.title("Graph of the function f(x) = (x - 3)^2 + 1") plt.xlabel('x') plt.ylabel('f(x)') plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) plt.legend()

Show the plot

plt.grid(True) plt.show()Here is the graph of the function $f(x) = (x - 3)^2 + 1$.

• The vertex of the parabola is at $(3, 1)$.
• The axis of symmetry is the vertical line $x = 3$.
• The y-intercept is at $(0, 10)$, where the curve crosses the y-axis.
• Since the coefficient of $(x - 3)^2$ is positive, the parabola opens upwards.

Would you like further details or have any specific questions on this graph?

Here are some related questions for further exploration:

1. How can you determine the domain and range of this function?
2. What would happen if the coefficient of $(x - 3)^2$ were negative?
3. How would the graph change if the constant $+1$ were replaced with another value?
4. What is the effect of shifting the vertex horizontally or vertically in quadratic functions?
5. How can you find the x-intercepts of a quadratic function like this one?

Tip: When graphing quadratic functions, always start by identifying the vertex and axis of symmetry, as they give you a lot of insight into the graph's shape and direction!