Key Features Of The Quadratic Equation Y=(x+12)(x+6)

This article delves into the key features of the quadratic equation $y = (x+12)(x+6)$, providing a comprehensive analysis of its intercepts, vertex, and overall shape. Understanding these features is crucial for graphing the equation and solving related problems in algebra and calculus. We will explore each feature in detail, ensuring a clear and thorough understanding of the equation's behavior. Quadratic equations are fundamental in mathematics, appearing in various applications from physics to engineering, making their comprehension essential for students and professionals alike.

X-Intercepts: Where the Parabola Crosses the X-Axis

The x-intercepts of a quadratic equation are the points where the parabola intersects the x-axis. These points are also known as the roots or zeros of the equation. To find the x-intercepts, we set $y = 0$ and solve for $x$. In the given equation, $y = (x+12)(x+6)$, setting $y$ to zero gives us $(x+12)(x+6) = 0$. This equation is satisfied when either $(x+12) = 0$ or $(x+6) = 0$.

Solving for $x$ in each case, we get:

• $x + 12 = 0 ightarrow x = -12$
• $x + 6 = 0 ightarrow x = -6$

Therefore, the x-intercepts are $(-12, 0)$ and $(-6, 0)$. These points are crucial as they define where the parabola crosses the x-axis, giving us a foundational understanding of the graph's position. The x-intercepts also help in determining the axis of symmetry, which passes through the midpoint of these two points. In this case, the axis of symmetry is $x = -9$, which we will discuss further in the vertex section. Understanding the x-intercepts is not just about finding the roots; it's about visualizing the parabola's interaction with the x-axis and using this information to sketch the graph accurately. Moreover, in real-world applications, the x-intercepts can represent crucial points, such as the time at which a projectile hits the ground or the points where a profit function equals zero. Therefore, a thorough grasp of how to find and interpret x-intercepts is essential for both theoretical and practical problem-solving in mathematics.

Y-Intercept: Where the Parabola Crosses the Y-Axis

The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, we set $x = 0$ in the equation and solve for $y$. In our equation, $y = (x+12)(x+6)$, substituting $x = 0$ gives us:

$y = (0+12)(0+6) = 12 imes 6 = 72$

Thus, the y-intercept is $(0, 72)$. This point is significant because it shows where the parabola begins its upward curve after reaching its minimum point (vertex). The y-intercept provides a vertical anchor for the graph, helping to visualize the parabola's position on the coordinate plane. In practical terms, the y-intercept can represent the initial value of a function, such as the starting height of an object or the initial cost of a project. Knowing the y-intercept allows us to understand the context of the quadratic equation better and to predict its behavior. For example, if the equation represents the trajectory of a ball, the y-intercept could represent the height from which the ball was initially thrown. Similarly, in business applications, the y-intercept could represent the fixed costs of production, which are incurred even when no units are produced. Therefore, the y-intercept is a vital feature of a quadratic equation, offering both a graphical reference point and a meaningful interpretation in real-world scenarios.

Vertex: The Minimum Point of the Parabola

The vertex of a parabola is the point where the parabola changes direction. For a quadratic equation in the form $y = ax^2 + bx + c$ where $a > 0$, the vertex represents the minimum point of the parabola. In our case, $y = (x+12)(x+6)$, we can expand the equation to get $y = x^2 + 18x + 72$. The x-coordinate of the vertex can be found using the formula x = -rac{b}{2a}.

In our expanded equation, $a = 1$ and $b = 18$, so the x-coordinate of the vertex is:

x = -rac{18}{2(1)} = -9

To find the y-coordinate of the vertex, we substitute $x = -9$ back into the original equation:

$y = (-9+12)(-9+6) = (3)(-3) = -9$

Therefore, the vertex is $(-9, -9)$. This point is the absolute minimum of the parabola, meaning it is the lowest point on the graph. The vertex is a critical feature because it determines the symmetry of the parabola; the vertical line passing through the vertex is the axis of symmetry. The axis of symmetry is equidistant from the x-intercepts, and in our case, it is the line $x = -9$. Understanding the vertex is essential for graphing the parabola accurately and for solving optimization problems. In practical applications, the vertex can represent the maximum or minimum value of a quantity, such as the maximum height of a projectile or the minimum cost of production. For instance, if the quadratic equation represents the profit function of a company, the vertex would indicate the production level that yields the highest profit. Therefore, the vertex is a central concept in the study of quadratic equations, providing valuable information about the parabola's shape, symmetry, and extreme values.

In summary, the key features of the equation $y = (x+12)(x+6)$ are:

• The x-intercepts are $(-12, 0)$ and $(-6, 0)$.
• The y-intercept is $(0, 72)$.
• The vertex is an absolute minimum at $(-9, -9)$.

These features provide a comprehensive understanding of the parabola's behavior and position on the coordinate plane. By identifying these key points, we can accurately graph the quadratic equation and solve related problems.

Detailed Explanation of Each Feature

To further elaborate on the key features, let's delve into a more detailed explanation of each, reinforcing the concepts and providing additional insights.

X-Intercepts in Depth

The x-intercepts, as mentioned earlier, are the points where the parabola intersects the x-axis. They are also known as the roots or zeros of the quadratic equation. Finding the x-intercepts involves setting $y = 0$ and solving for $x$. The equation $y = (x+12)(x+6)$ is already in factored form, which makes it particularly easy to find the x-intercepts. The factored form of a quadratic equation is $y = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots or x-intercepts. In our equation, we can see that $r_1 = -12$ and $r_2 = -6$.

The x-intercepts are crucial for several reasons. First, they give us the points where the parabola crosses the x-axis, providing a visual anchor for the graph. Second, they help us determine the interval(s) where the function is positive or negative. For example, in our case, the parabola is below the x-axis (i.e., $y < 0$) between $x = -12$ and $x = -6$, and above the x-axis (i.e., $y > 0$) for $x < -12$ and $x > -6$. This information is invaluable for solving inequalities involving quadratic expressions. Furthermore, the x-intercepts are directly related to the solutions of the quadratic equation when set equal to zero. The quadratic formula, x = rac{-b extbackslashpm \sqrt{b^2 - 4ac}}{2a}, provides a general method for finding the x-intercepts of any quadratic equation in the form $y = ax^2 + bx + c$. However, when the equation is in factored form, as in our case, we can easily find the x-intercepts by setting each factor equal to zero. The x-intercepts also play a significant role in real-world applications. For instance, in physics, they can represent the time at which a projectile lands on the ground, and in business, they can represent the break-even points where a company's revenue equals its costs. Therefore, understanding x-intercepts is fundamental to both the theoretical understanding and practical application of quadratic equations.

Y-Intercept Elaboration

The y-intercept is the point where the parabola intersects the y-axis. As discussed earlier, we find the y-intercept by setting $x = 0$ in the equation and solving for $y$. For the equation $y = (x+12)(x+6)$, setting $x = 0$ gives us $y = (0+12)(0+6) = 72$, so the y-intercept is $(0, 72)$. The y-intercept provides important information about the parabola's vertical position on the coordinate plane. It is the point where the parabola starts its upward curve after reaching its minimum point (the vertex). The y-intercept is also the value of the function when $x = 0$, which can have significant meaning in various contexts.

For example, if the quadratic equation represents the height of an object thrown into the air as a function of time, the y-intercept would represent the initial height of the object when it was thrown. Similarly, in business applications, the y-intercept could represent the fixed costs of a company, which are the costs that do not vary with the level of production. The y-intercept can also help us determine the scale of the y-axis when graphing the parabola. In our case, the y-intercept is 72, which tells us that the y-axis needs to extend at least up to 72 to accurately represent the graph. Understanding the y-intercept is crucial for interpreting the behavior of the quadratic function and for applying it to real-world scenarios. It provides a starting point for understanding the function's overall trend and can be used to make predictions about its behavior. In addition, the y-intercept, along with the x-intercepts and vertex, helps us sketch the graph of the parabola accurately, providing a complete picture of the quadratic function's behavior.

Vertex Deep Dive

The vertex, as the turning point of the parabola, is arguably one of its most significant features. For a quadratic equation in the form $y = ax^2 + bx + c$, the vertex represents the minimum point if $a > 0$ and the maximum point if $a < 0$. In our equation, $y = (x+12)(x+6)$, which expands to $y = x^2 + 18x + 72$, we have $a = 1$, so the vertex is a minimum point. The x-coordinate of the vertex can be found using the formula x = -rac{b}{2a}. In our case, $a = 1$ and $b = 18$, so the x-coordinate of the vertex is x = -rac{18}{2(1)} = -9.

To find the y-coordinate, we substitute $x = -9$ back into the original equation: $y = (-9+12)(-9+6) = (3)(-3) = -9$. Thus, the vertex is $(-9, -9)$. The vertex is the point of symmetry for the parabola; the vertical line passing through the vertex, known as the axis of symmetry, divides the parabola into two mirror-image halves. The axis of symmetry is given by the equation $x = -9$ in our case. Understanding the vertex is crucial for graphing the parabola accurately. It provides the lowest point (or highest point if $a < 0$) on the graph, and it helps us understand the parabola's overall shape and direction. The vertex also has significant applications in optimization problems. For example, if the quadratic equation represents the profit function of a company, the vertex would indicate the production level that yields the highest profit. Similarly, if the equation represents the height of a projectile, the vertex would indicate the maximum height reached by the projectile. The vertex is also related to the range of the quadratic function. Since the vertex is the minimum point in our case, the range of the function is $y ge -9$, meaning that the y-values of the function are always greater than or equal to -9. Therefore, a thorough understanding of the vertex is essential for solving a wide range of problems involving quadratic equations and for applying them to real-world situations.

In conclusion, understanding the key features of the quadratic equation $y = (x+12)(x+6)$—the x-intercepts, y-intercept, and vertex—is essential for analyzing and graphing the parabola. These features provide valuable insights into the equation's behavior and its applications in various fields. By mastering these concepts, one can confidently tackle quadratic equation problems and appreciate their significance in mathematics and beyond.