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

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!

Learn how to graph the quadratic function f(x) = (x - 3)^2 + 1. This problem covers the vertex form of a quadratic, key graph features such as the vertex, axis of symmetry, y-intercept, and the shape of the parabola. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation