Finding X And Y Intercepts Of Y=x^2+x-6 A Step-by-Step Guide

Finding the $x$- and $y$-intercepts of a graph is a fundamental concept in algebra and is crucial for understanding the behavior of functions. In this comprehensive guide, we will delve into the process of determining the intercepts for the given quadratic function $y = x^2 + x - 6$. We'll explore the underlying principles, step-by-step calculations, and graphical interpretations to provide a clear and thorough understanding. Mastering this skill is essential for various mathematical applications, including graphing functions, solving equations, and analyzing real-world problems modeled by quadratic relationships.

Understanding Intercepts

Before diving into the calculations, it's important to define what $x$- and $y$-intercepts represent. The x-intercepts are the points where the graph of the function intersects the $x$-axis. At these points, the $y$-coordinate is always zero. Conversely, the y-intercept is the point where the graph intersects the $y$-axis, and at this point, the $x$-coordinate is zero. These intercepts provide key insights into the function's behavior and its relationship with the coordinate axes. They serve as anchor points for sketching the graph and can help in identifying important features such as the vertex and axis of symmetry for quadratic functions. Understanding intercepts is not just a mathematical exercise; it's a fundamental skill that enables us to visualize and interpret mathematical relationships in a graphical context.

Finding the $x$-intercepts

To find the $x$-intercepts, we set $y = 0$ in the equation $y = x^2 + x - 6$ and solve for $x$. This is because, as mentioned earlier, the $y$-coordinate is zero at the points where the graph intersects the $x$-axis. The resulting equation is a quadratic equation: $x^2 + x - 6 = 0$. Solving quadratic equations is a core skill in algebra, and there are several methods to choose from, including factoring, completing the square, and using the quadratic formula. In this case, factoring is the most straightforward approach. We look for two numbers that multiply to -6 and add to 1 (the coefficient of the $x$ term). These numbers are 3 and -2. Therefore, we can factor the quadratic equation as $(x + 3)(x - 2) = 0$. Setting each factor equal to zero gives us two solutions: $x + 3 = 0$ which implies $x = -3$, and $x - 2 = 0$ which implies $x = 2$. These solutions represent the $x$-coordinates of the $x$-intercepts. The $x$-intercepts are the points where the graph crosses the $x$-axis, and in this case, they are $(-3, 0)$ and $(2, 0)$. These points are crucial for sketching the parabola and understanding its position relative to the $x$-axis.

Step-by-step factoring

The quadratic equation we need to solve is $x^2 + x - 6 = 0$. Factoring this equation involves finding two binomials that, when multiplied together, give us the original quadratic expression. Here's a step-by-step breakdown of the factoring process:

1. Identify the coefficients: In the equation $x^2 + x - 6 = 0$, the coefficient of the $x^2$ term is 1, the coefficient of the $x$ term is 1, and the constant term is -6.
2. Find two numbers: We need to find two numbers that multiply to the constant term (-6) and add up to the coefficient of the $x$ term (1). These two numbers are 3 and -2 because $3 imes -2 = -6$ and $3 + (-2) = 1$.
3. Write the factored form: Using these two numbers, we can write the quadratic expression in factored form as $(x + 3)(x - 2)$.
4. Verify the factoring: To verify that the factoring is correct, we can multiply the two binomials together: $(x + 3)(x - 2) = x(x - 2) + 3(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6$ This confirms that our factored form is correct.
5. Set each factor to zero: Now we set each factor equal to zero and solve for $x$:
   • $x + 3 = 0 ightarrow x = -3$
   • $x - 2 = 0 ightarrow x = 2$

Thus, the solutions to the quadratic equation are $x = -3$ and $x = 2$, which correspond to the $x$-coordinates of the $x$-intercepts.

Graphical Interpretation of $x$-intercepts

Graphically, the $x$-intercepts are the points where the parabola intersects the $x$-axis. For the equation $y = x^2 + x - 6$, the $x$-intercepts are at $x = -3$ and $x = 2$. This means the parabola crosses the $x$-axis at these two points. The $x$-intercepts are crucial for understanding the behavior of the quadratic function because they divide the $x$-axis into intervals where the function's values are either positive or negative. Between the $x$-intercepts (-3 and 2), the parabola lies below the $x$-axis, indicating that the function's values are negative in this interval. Outside this interval, the parabola lies above the $x$-axis, meaning the function's values are positive. This graphical interpretation provides a visual understanding of the function's roots and its overall shape. Moreover, the $x$-intercepts, along with the vertex, help in accurately sketching the graph of the parabola.

Finding the $y$-intercept

To find the $y$-intercept, we set $x = 0$ in the equation $y = x^2 + x - 6$ and solve for $y$. This is because the $x$-coordinate is zero at the point where the graph intersects the $y$-axis. Substituting $x = 0$ into the equation, we get $y = (0)^2 + (0) - 6$, which simplifies to $y = -6$. Therefore, the $y$-intercept is the point $(0, -6)$. The $y$-intercept is another key point on the graph of the function. It indicates where the parabola crosses the $y$-axis and provides a reference point for understanding the vertical position of the graph. In this case, the $y$-intercept is at $(0, -6)$, which means the parabola intersects the $y$-axis six units below the origin. The $y$-intercept, along with the $x$-intercepts, helps in visualizing the overall shape and position of the parabola on the coordinate plane. It also plays a role in determining the range of the function and its minimum or maximum value.

Practical Significance of Intercepts

Intercepts are not just theoretical points; they often have significant practical interpretations in real-world applications. For example, if the equation represents the trajectory of a projectile, the $x$-intercepts could represent the points where the projectile lands, and the $y$-intercept could represent the initial height from which the projectile was launched. In business, if the equation represents a profit function, the $x$-intercepts could represent the break-even points (where profit is zero), and the $y$-intercept could represent the initial costs or losses. Understanding the intercepts provides valuable information about the scenario being modeled by the function. They allow us to make predictions, analyze trends, and solve practical problems. For instance, in a supply and demand model, the intercepts can represent the price at which there is no demand or the quantity supplied when the price is zero. Therefore, the ability to find and interpret intercepts is a crucial skill in applying mathematical concepts to real-world situations.

Summarizing the Results

In summary, to find the $x$-intercepts of the graph of $y = x^2 + x - 6$, we set $y = 0$ and solved the resulting quadratic equation $x^2 + x - 6 = 0$. The factored form of the equation is $(x + 3)(x - 2) = 0$, which gives us the solutions $x = -3$ and $x = 2$. Thus, the $x$-intercepts are $(-3, 0)$ and $(2, 0)$. To find the $y$-intercept, we set $x = 0$ in the original equation, which gives us $y = (0)^2 + (0) - 6 = -6$. Therefore, the $y$-intercept is $(0, -6)$. These intercepts provide key points for graphing the quadratic function and understanding its behavior. The $x$-intercepts indicate where the parabola crosses the $x$-axis, and the $y$-intercept indicates where it crosses the $y$-axis. Together, these intercepts, along with the vertex and axis of symmetry, provide a comprehensive picture of the parabola's shape and position on the coordinate plane. Mastering the process of finding intercepts is essential for analyzing and interpreting quadratic functions and their applications.

Conclusion

Finding the $x$- and $y$-intercepts of a quadratic function is a fundamental skill in algebra with significant applications in various fields. By setting $y = 0$ and solving for $x$, we can determine the $x$-intercepts, which are the points where the graph intersects the $x$-axis. Similarly, by setting $x = 0$ and solving for $y$, we can find the $y$-intercept, which is the point where the graph intersects the $y$-axis. For the given equation $y = x^2 + x - 6$, the $x$-intercepts are $(-3, 0)$ and $(2, 0)$, and the $y$-intercept is $(0, -6)$. These intercepts provide crucial information about the function's behavior and are essential for graphing and analyzing quadratic relationships. Understanding intercepts not only enhances mathematical proficiency but also provides valuable insights into real-world scenarios modeled by quadratic functions. From projectile motion to business profit analysis, the ability to find and interpret intercepts is a powerful tool for problem-solving and decision-making.