To solve the quadratic equation $y = x^2 - x - 12$ using the graphical method, follow these steps:

1. Identify the quadratic function:

The equation represents a parabola: $y = x^2 - x - 12$

2. Find the x-intercepts (roots):

The x-intercepts occur when $y = 0$, meaning we need to solve: $x^2 - x - 12 = 0$

To solve by factoring: $x^2 - x - 12 = (x - 4)(x + 3) = 0$

Thus, the solutions are: $x = 4 \quad \text{and} \quad x = -3$ So, the parabola intersects the x-axis at $x = 4$ and $x = -3$.

3. Find the vertex:

The vertex of the parabola occurs at the axis of symmetry, given by the formula: $x = \frac{-b}{2a}$ For the equation $y = ax^2 + bx + c$, here $a = 1$, $b = -1$, and $c = -12$: $x = \frac{-(-1)}{2(1)} = \frac{1}{2} = 0.5$ Substitute $x = 0.5$ into the equation to find the y-coordinate of the vertex: $y = (0.5)^2 - 0.5 - 12 = 0.25 - 0.5 - 12 = -12.25$ Thus, the vertex is at $(0.5, -12.25)$.

4. Plot the parabola:

• The parabola opens upwards because the coefficient of $x^2$ is positive.
• Plot the x-intercepts at $x = 4$ and $x = -3$.
• Plot the vertex at $(0.5, -12.25)$.
• Sketch the curve through these points, ensuring the parabola is symmetric around the vertex.

5. Interpret the roots:

From the graph, the x-intercepts $x = 4$ and $x = -3$ are the solutions to the quadratic equation.

Do you want details on how to graph this using software or further elaboration?

Here are 5 related questions:

1. What are other methods to solve quadratics besides graphical methods?
2. How does the vertex form of a quadratic equation relate to its graph?
3. How do you determine whether a parabola opens upwards or downwards?
4. What is the significance of the discriminant in quadratic equations?
5. Can all quadratic equations be factored easily?

Tip: To confirm the accuracy of a graph, always verify the roots algebraically.