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

1. Rewrite the equation as a function:
   $y = x^2 - 2x - 3$

2. Graph the function:
   Plot the parabola described by the equation $y = x^2 - 2x - 3$ on the coordinate plane.

3. Identify the x-intercepts:
   The solutions to the equation $x^2 - 2x - 3 = 0$ are the x-values where the graph intersects the x-axis (i.e., where $y = 0$).

Steps:

• Step 1: Identify key points like the vertex and the axis of symmetry.
  The axis of symmetry is given by $x = -\frac{b}{2a}$. Here, $a = 1$ and $b = -2$, so: $x = \frac{-(-2)}{2(1)} = 1$ The vertex lies on this line.

• Step 2: Compute the vertex by substituting $x = 1$ into the equation: $y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$ The vertex is at $(1, -4)$.

• Step 3: Plot additional points to create the shape of the parabola.

• Step 4: Draw the graph and identify where it intersects the x-axis.

When you graph the parabola, you will see that it intersects the x-axis at two points, which are the solutions to the equation.

Graph:

The graph of $y = x^2 - 2x - 3$ will intersect the x-axis at $x = -1$ and $x = 3$.

Solution:

The solutions to the equation $x^2 - 2x - 3 = 0$ are $x = -1$ and $x = 3$.

Would you like more details or have any questions?

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Related Questions:

1. How do you find the vertex of a parabola from its equation?
2. How do you solve quadratic equations using the quadratic formula?
3. What is the significance of the discriminant in a quadratic equation?
4. How do you determine the direction of a parabola (whether it opens upwards or downwards)?
5. How do you factor quadratic equations to find their roots?

Tip: When graphing a quadratic equation, always start by finding the vertex and axis of symmetry to make plotting easier.