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

1. Rewrite the equation in standard form:

The given equation is already in the form: $y = x^2 - 2x - 3$ This is the equation of a parabola.

2. Plot the parabola:

To graph the quadratic function, we will calculate a few values of $x$ and the corresponding values of $y = x^2 - 2x - 3$, then plot those points.

3. Identify the x-intercepts:

The x-intercepts of the graph (where the graph crosses the x-axis) are the solutions of the equation $x^2 - 2x - 3 = 0$. These intercepts are the points where $y = 0$.

Let’s start by factoring the quadratic equation: $x^2 - 2x - 3 = (x - 3)(x + 1) = 0$ So, the solutions are: $x = 3 \quad \text{or} \quad x = -1$ These are the x-intercepts, meaning the graph crosses the x-axis at $x = 3$ and $x = -1$.

4. Sketch the graph:

• The parabola opens upwards since the coefficient of $x^2$ is positive.
• The x-intercepts are at $x = 3$ and $x = -1$.
• The vertex can be found using the formula for the x-coordinate of the vertex, $x = -\frac{b}{2a}$. For this equation, $a = 1$ and $b = -2$: $x = -\frac{-2}{2(1)} = 1$ Substituting $x = 1$ into the equation to find the y-coordinate of the vertex: $y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4$ So, the vertex is at $(1, -4)$.

By plotting the vertex and the x-intercepts, the parabola will have its minimum point at $(1, -4)$, and it will pass through $(-1, 0)$ and $(3, 0)$.

Thus, the solution to the equation $x^2 - 2x - 3 = 0$ by graphing is $x = 3$ and $x = -1$.

Would you like a graph for better visualization or further details?

5 Questions to Explore Further:

1. How do you determine the vertex of a quadratic equation?
2. What is the significance of the x-intercepts in a quadratic equation?
3. Can all quadratic equations be factored easily? If not, what methods can be used to solve them?
4. What does the direction of a parabola (opening upwards or downwards) depend on?
5. How can the quadratic formula be used to solve quadratic equations?

Tip:

The vertex of a parabola gives you the minimum or maximum value of the quadratic function, depending on whether the parabola opens upwards or downwards.