The quadratic parent function is:
$f(x) = x^2$

Key Features of the Quadratic Parent Function:

• It is a parabola that opens upward.
• The vertex is at (0,0).
• It is symmetric about the y-axis.
• The standard five points used to graph it are:
  • Vertex: $(0,0)$
  • Two points on the left: $(-1,1)$ and $(-2,4)$
  • Two points on the right: $(1,1)$ and $(2,4)$

Graphing the Function

I will generate a graph with these five points clearly marked. numpy as np import matplotlib.pyplot as plt

Define the quadratic function

def quadratic(x): return x**2

Generate x values

x_values = np.linspace(-3, 3, 100) y_values = quadratic(x_values)

Key points

key_x = np.array([-2, -1, 0, 1, 2]) key_y = quadratic(key_x)

Plot the function

plt.figure(figsize=(6,6)) plt.plot(x_values, y_values, label=r"$f(x) = x^2$", color="blue") plt.scatter(key_x, key_y, color="red", zorder=3, label="Key Points")

Labels and grid

plt.xlabel("x-axis") plt.ylabel("y-axis") plt.title("Quadratic Parent Function") plt.axhline(0, color="black", linewidth=0.8) plt.axvline(0, color="black", linewidth=0.8) plt.grid(True, linestyle="--", alpha=0.6)

Annotate key points

for (x, y) in zip(key_x, key_y): plt.text(x, y, f" ({x},{y})", fontsize=12, verticalalignment='bottom')

plt.legend() plt.show()Here is the graph of the quadratic parent function $f(x) = x^2$ with five clearly marked points:
$(-2,4), (-1,1), (0,0), (1,1), (2,4)$ Let me know if you have any questions or need further details!

Related Questions:

1. What are the transformations of the quadratic function?
2. How does changing the coefficient of $x^2$ affect the graph?
3. What is the axis of symmetry for a quadratic function?
4. How do you find the vertex of any quadratic function?
5. What are the x-intercepts of a quadratic function and how do you find them?

Tip:

To quickly graph a quadratic function, identify its vertex, symmetry, and a few key points, then reflect them across the axis of symmetry!