To solve the quadratic equation $x^2 - 2x - 3 = 0$ by graphing, follow these steps:

1. Write the equation in standard form:

$y = x^2 - 2x - 3$

2. Identify key features of the quadratic equation:

• Vertex: The vertex can be found using $x = -\frac{b}{2a}$, where $a = 1$ and $b = -2$. Thus, $x = -\frac{-2}{2(1)} = 1.$ Plugging $x = 1$ into the equation gives: $y = (1)^2 - 2(1) - 3 = -4.$ So the vertex is $(1, -4)$.

• Intercepts:

  • y-intercept: At $x = 0$, $y = 0^2 - 2(0) - 3 = -3.$ So the y-intercept is $(0, -3)$.
  • x-intercepts (solutions): These are the values of $x$ when $y = 0$, which are the solutions to $x^2 - 2x - 3 = 0$. Let's find them graphically.

3. Plot the quadratic function:

The graph is a parabola opening upwards because the coefficient of $x^2$ ($a = 1$) is positive.

I will now graph this function and identify where it crosses the x-axis.import numpy as np import matplotlib.pyplot as plt

Define the quadratic function

def quadratic(x): return x**2 - 2*x - 3

Generate x values for the graph

x_vals = np.linspace(-2, 4, 500) y_vals = quadratic(x_vals)

Plot the quadratic function

plt.figure(figsize=(8, 6)) plt.plot(x_vals, y_vals, label=r"$y = x^2 - 2x - 3$", linewidth=2)

Highlight the x-intercepts (roots)

roots = [(-1, quadratic(-1)), (3, quadratic(3))] # Approximate solutions from factorization for root in roots: plt.scatter(root[0], root[1], color='red', label=f"Root at x = {root[0]}")

Highlight vertex and y-intercept

plt.scatter(1, -4, color='green', label="Vertex (1, -4)") plt.scatter(0, -3, color='blue', label="y-Intercept (0, -3)")

Add labels, title, and legend

plt.axhline(0, color='black',linewidth=0.7, linestyle="--") plt.axvline(0, color='black',linewidth=0.7, linestyle="--") plt.title("Graph of $y = x^2 - 2x - 3$", fontsize=16) plt.xlabel("x", fontsize=14) plt.ylabel("y", fontsize=14) plt.grid(alpha=0.3) plt.legend(fontsize=12) plt.show() <<ImageDisplayed>>successFrom the graph, the x-intercepts (roots) of the equation $x^2 - 2x - 3 = 0$ are clearly visible at $x = -1$ and $x = 3$. These are the solutions to the equation.

Would you like a detailed explanation of the graph or further assistance?

Related Questions:

1. How can we confirm the roots algebraically?
2. What is the significance of the vertex in this quadratic equation?
3. How can the quadratic formula be applied to solve this equation?
4. What role do the intercepts play in graphing the quadratic function?
5. How can transformations of the parent function $y = x^2$ relate to this graph?

Tip:

When graphing quadratic equations, always check the vertex, axis of symmetry, and intercepts to ensure an accurate representation!