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.

y = x² - x - 12 solve by graphical method

Learn how to solve the quadratic equation y = x² - x - 12 using the graphical method. This step-by-step guide covers algebraic concepts such as factoring, finding x-intercepts, and determining the vertex of a parabola. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation