Zeros And Vertex Of H(r) = (r+1)(r+8)

Hey guys! Let's dive into the fascinating world of quadratic functions, and specifically, we're going to dissect the function h(r) = (r+1)(r+8). We'll be figuring out its zeros and pinpointing the vertex of its parabolic graph. Buckle up, because math can be an exciting adventure!

1) Discovering the Zeros of the Function

When we talk about the zeros of a function, we're essentially asking: "For what values of 'r' does the function h(r) equal zero?" In other words, we're looking for the points where the parabola intersects the x-axis (or the r-axis in this case). This is a fundamental concept in understanding the behavior of any function, especially quadratic functions which have a distinctive U-shaped curve. Finding these zeros is crucial for a multitude of applications, ranging from physics simulations where we might be modeling projectile trajectories, to engineering designs where we might be optimizing structural stability. Moreover, understanding the zeros of a function helps us to visualize the function's graph more accurately, providing insights into its increasing and decreasing intervals, and its overall symmetry.

Our function, h(r) = (r+1)(r+8), is conveniently presented in its factored form. This is a huge advantage because it makes finding the zeros incredibly straightforward. Remember the zero-product property: if the product of two factors is zero, then at least one of the factors must be zero. Applying this to our function, we can set each factor equal to zero and solve for 'r'. So, we have two simple equations to solve:

• r + 1 = 0
• r + 8 = 0

Solving the first equation, we subtract 1 from both sides, giving us r = -1. Similarly, solving the second equation, we subtract 8 from both sides, resulting in r = -8. These are our zeros! They represent the r-values where the parabola crosses the r-axis. But wait, there's a slight twist in the question! It asks us to write the smaller r first, and then the larger r. Comparing -1 and -8, we know that -8 is smaller than -1 (remember, on the number line, numbers become smaller as we move to the left). Therefore, the smaller zero is -8, and the larger zero is -1.

So, to recap, finding the zeros of a quadratic function is like uncovering the hidden roots that anchor its graph. In the context of our function, h(r) = (r+1)(r+8), the factored form provided us with a direct pathway to these roots, simplifying the process significantly. Understanding the significance of the zeros extends beyond mere mathematical calculations; it allows us to interpret the function's behavior in real-world scenarios, making it an essential tool in various fields of study and application. The zeros, in essence, define the points of equilibrium, transition, or critical change within the system that the function models. This is why the ability to accurately determine the zeros of a function is a cornerstone of mathematical literacy.

2) Pinpointing the Vertex of the Parabola

Now, let's shift our focus to another crucial feature of the parabola: the vertex. The vertex is the turning point of the parabola; it's either the lowest point (for parabolas that open upwards) or the highest point (for parabolas that open downwards). It represents the minimum or maximum value of the function, respectively. Finding the vertex is key to understanding the parabola's overall shape and behavior. In practical terms, the vertex can represent a point of optimization – for example, the maximum height of a projectile's trajectory or the minimum cost in a production process. It's a point of significant interest in many applications.

There are several ways to find the vertex of a parabola. One common method involves using the vertex formula, which is derived from the standard form of a quadratic equation. However, since our function is given in factored form, there's a slightly more elegant approach we can take. We can leverage the symmetry of the parabola. Parabolas are symmetrical about a vertical line that passes through the vertex. This line is called the axis of symmetry. The x-coordinate (or r-coordinate in our case) of the vertex lies exactly midway between the two zeros of the function.

We already found the zeros in the previous section: r = -8 and r = -1. To find the r-coordinate of the vertex, we simply calculate the average of these two values:

r-coordinate of vertex = (-8 + (-1)) / 2 = -9 / 2 = -4.5

So, the r-coordinate of our vertex is -4.5. Now, to find the corresponding h(r)-coordinate (the y-coordinate, in more general terms), we substitute this value back into our original function:

h(-4.5) = (-4.5 + 1)(-4.5 + 8) = (-3.5)(3.5) = -12.25

Therefore, the vertex of the parabola is (-4.5, -12.25). This point represents the minimum value of our function, as the coefficient of the r² term in the expanded form of the function is positive (which means the parabola opens upwards). The ability to accurately locate the vertex is essential not only for sketching the graph of the function but also for solving optimization problems in various fields such as physics, engineering, and economics. Understanding the vertex helps us grasp the function's extreme values and the conditions under which they occur.

In summary, pinpointing the vertex of a parabola unveils its most crucial characteristic: the point of maximum or minimum value. By utilizing the symmetry inherent in parabolas and leveraging the zeros we previously calculated, we efficiently determined the vertex of our function, h(r) = (r+1)(r+8). This process underscores the interconnectedness of different mathematical concepts and their practical implications in real-world applications. The vertex, therefore, serves as a critical anchor point in our understanding of the function's behavior and its potential applications.

In conclusion, we've successfully navigated the landscape of our quadratic function, uncovering its zeros and pinpointing its vertex. These key features paint a comprehensive picture of the parabola's shape and behavior, highlighting the power of mathematical analysis in understanding complex functions. This process is fundamental not only in academic mathematics but also in a myriad of real-world applications where optimization and understanding extreme values are critical. Keep exploring, guys, and you'll find that math is full of exciting discoveries!