Yes! The given points describe a quadratic function:

• Vertex: $(2.5, -2.25)$
• X-intercepts: $(1,0)$ and $(4,0)$
• Y-intercept: $(0,4)$

Since the x-intercepts are known, the quadratic function can be written in factored form:
$y = a(x - 1)(x - 4)$ To find $a$, substitute the y-intercept $(0,4)$:

$4 = a(0 - 1)(0 - 4)$

$4 = a(4)$

$a = -1$

So the quadratic equation is:

$y = -1(x - 1)(x - 4)$

Now, I'll generate a sketch of the graph. numpy as np import matplotlib.pyplot as plt

Define the quadratic function

def quadratic(x): return -1 * (x - 1) * (x - 4)

Generate x values

x_values = np.linspace(-1, 6, 400) y_values = quadratic(x_values)

Plot the function

plt.figure(figsize=(6,6)) plt.plot(x_values, y_values, label=r'$y = -(x - 1)(x - 4)$', color='blue')

Plot the key points

points = {'Vertex (2.5, -2.25)': (2.5, -2.25), 'X-Intercept (1,0)': (1,0), 'X-Intercept (4,0)': (4,0), 'Y-Intercept (0,4)': (0,4)}

for label, (x, y) in points.items(): plt.scatter(x, y, color='red', zorder=3) plt.annotate(label, (x, y), textcoords="offset points", xytext=(5,-10), ha='left', fontsize=10)

Axis settings

plt.axhline(0, color='black', linewidth=1) # x-axis plt.axvline(0, color='black', linewidth=1) # y-axis plt.grid(True, linestyle='--', alpha=0.5) plt.xlabel("x") plt.ylabel("y") plt.title("Quadratic Function Graph") plt.legend() plt.show()Here is the graph of the quadratic function $y = -(x - 1)(x - 4)$. It passes through the given vertex, x-intercepts, and y-intercept.

Would you like a step-by-step explanation of how this was constructed?

Here are 5 related questions you might find interesting:

1. How do you find the equation of a quadratic function given three points?
2. How does the value of $a$ in a quadratic function affect the graph's shape?
3. Can you determine the axis of symmetry from the equation of a quadratic function?
4. What are other methods to find the vertex of a quadratic function?
5. How do transformations affect the standard quadratic function $y = ax^2 + bx + c$?

Tip: The axis of symmetry of a quadratic function in standard form $y = ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$.