Exploring F(x) = X^2 - 9 Completing The Table And Graphing

Hey guys! Today, we're diving into the fascinating world of quadratic functions, specifically the function f(x) = x^2 - 9. We're going to explore this function by completing a table of values, which will give us a better understanding of its behavior and graph. So, grab your calculators and let's get started!

Understanding the Function f(x) = x^2 - 9

At its heart, understanding quadratic functions like f(x) = x^2 - 9 involves recognizing its fundamental form and how the different components shape its graph. This particular function is a classic example of a parabola, a U-shaped curve that opens upwards because the coefficient of the x^2 term is positive (in this case, 1). The “-9” part of the equation plays a crucial role; it shifts the entire parabola downwards by 9 units compared to the basic function f(x) = x^2. This vertical shift significantly impacts where the parabola intersects the y-axis, making it cross at the point (0, -9).

To truly grasp the function's behavior, it's essential to consider how different x-values are transformed into y-values. When x is 0, y becomes -9, marking the vertex (or the lowest point) of the parabola. As x moves away from 0 in either direction, whether positive or negative, the x^2 term increases, causing y to rise. This symmetrical increase around the vertex is a key characteristic of quadratic functions. Understanding this symmetry and the vertical shift helps us predict the general shape and position of the parabola on the coordinate plane. Moreover, recognizing these features allows us to quickly sketch the graph and identify key points such as the vertex and intercepts, making it easier to analyze the function's properties and solve related problems.

Completing the Table of Values

Completing the table of values is a fantastic way to visualize how a function behaves for different inputs. We're given the function f(x) = x^2 - 9 and a partially filled table. Our mission, should we choose to accept it (and we do!), is to find the missing y-values for the given x-values and vice-versa. This process is all about substituting the given x-values into the function and calculating the corresponding y-values, and conversely, solving for x when y-values are provided.

Let's start with the easy ones. When x = -2, we plug it into the function: f(-2) = (-2)^2 - 9 = 4 - 9 = -5. So, when x is -2, y is -5. Next up, x = -1: f(-1) = (-1)^2 - 9 = 1 - 9 = -8. We're on a roll! Now, for x = 0: f(0) = (0)^2 - 9 = 0 - 9 = -9. These calculations are building a picture of the parabola's shape around its vertex.

But what about the other way around? We're given y-values and need to find the corresponding x-values. When y = 16, we set up the equation 16 = x^2 - 9. Adding 9 to both sides gives us 25 = x^2. Taking the square root of both sides, we get x = ±5. Remember, square roots can be positive or negative! This gives us two points on the parabola. Finally, when y = 27, we have 27 = x^2 - 9. Adding 9 to both sides gives us 36 = x^2. Taking the square root, we get x = ±6. By methodically filling in these values, we gain a concrete understanding of how the function transforms x-values into y-values, and we're one step closer to sketching its graph.

Step-by-Step Calculation of Missing Values

Let's break down the step-by-step calculation of those missing values. We've already touched on the process, but let's really drill down and make sure we've got this nailed. This isn't just about getting the right answers; it's about understanding the mechanics of how the function operates.

First, let's revisit finding y-values when x-values are given. Imagine you're a little calculator, and x is the number being fed into you. You square it, then subtract 9. Simple as that! For instance, when x = -2, your internal workings would go like this: (-2)^2 = 4, then 4 - 9 = -5. Voila, y = -5. We repeat this process for each given x-value. This direct substitution is a fundamental skill in evaluating functions.

Now, for the slightly trickier part: finding x-values when y-values are given. This involves a bit of algebraic maneuvering. We're essentially solving an equation. Remember the equation is f(x) = x^2 - 9, which we can rewrite as y = x^2 - 9. So, if we're given y = 16, we substitute that in: 16 = x^2 - 9. The goal is to isolate x. We add 9 to both sides, giving us 25 = x^2. Here's the crucial part: we take the square root of both sides. But remember, the square root of a number has two possible solutions: a positive and a negative one. So, √25 = ±5. This tells us that x could be either 5 or -5. We repeat this process for each given y-value. By understanding these step-by-step calculations, we not only complete the table but also solidify our understanding of how to work with functions algebraically.

The Completed Table

Graphing the Function

Graphing the function f(x) = x^2 - 9 using the completed table of values is like connecting the dots to reveal a beautiful picture. Each (x, y) pair in the table represents a point on the graph, and plotting these points gives us a visual representation of the function's behavior. This is where the magic happens, where algebra transforms into geometry, and we can truly see the parabola come to life.

First, let's think about what we know. We've got a series of points: (-2, -5), (-1, -8), (0, -9), (5, 16), (-5, 16), (6, 27), and (-6, 27). These are our anchors, the points that will guide us in drawing the curve. The more points we plot, the more accurate our graph will be. But even with just a few points, we can start to see the characteristic U-shape of the parabola forming.

We know that the vertex, the lowest point of the parabola, is at (0, -9). This is a critical point because it's the turning point of the curve. From the vertex, the parabola curves upwards symmetrically on both sides. The points (-1, -8) and (-2, -5) help us trace the left side of the parabola, while their counterparts (1, -8) and (2, -5) (which we could calculate if we wanted even more points) would mirror that shape on the right side. The points (5, 16) and (-5, 16) are further out, showing us how quickly the parabola rises as we move away from the vertex. And finally, the points (6, 27) and (-6, 27) give us an even clearer picture of the upward trajectory.

When we connect these points with a smooth curve, we see the full parabola. It's a symmetrical, U-shaped curve that opens upwards, with its lowest point at (0, -9). The graph visually confirms what we've learned algebraically: that f(x) = x^2 - 9 is a parabola shifted downwards by 9 units compared to the basic f(x) = x^2. Graphing not only solidifies our understanding but also provides a powerful visual tool for analyzing the function's behavior.

Key Features of the Graph

Key features of the graph of f(x) = x^2 - 9 are like the landmarks on a map, guiding us to understand the function's characteristics. Identifying these features gives us a deeper insight into how the function behaves and its relationship to the equation.

The most prominent feature is undoubtedly the vertex. As we've discussed, the vertex is the turning point of the parabola, the point where it changes direction. In this case, the vertex is at (0, -9). This point is significant because it's the minimum value of the function. Since the parabola opens upwards, the y-value of the vertex, -9, is the lowest possible y-value the function can have.

Next, let's consider the intercepts. Intercepts are the points where the graph crosses the x-axis and y-axis. The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. We already know that when x = 0, y = -9, so the y-intercept is (0, -9). Notice that in this case, the vertex and the y-intercept are the same point. The x-intercepts are the points where the graph crosses the x-axis, which occur when y = 0. To find these, we set f(x) = 0 and solve for x: 0 = x^2 - 9. Adding 9 to both sides gives us 9 = x^2. Taking the square root, we get x = ±3. So, the x-intercepts are (-3, 0) and (3, 0).

Another crucial feature is the axis of symmetry. This is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. For f(x) = x^2 - 9, the axis of symmetry is the line x = 0 (the y-axis). This symmetry is a fundamental characteristic of parabolas, and it means that for every point on one side of the axis of symmetry, there's a corresponding point on the other side.

By identifying these key features – the vertex, intercepts, and axis of symmetry – we gain a comprehensive understanding of the graph and the behavior of the quadratic function f(x) = x^2 - 9.

Applications of Quadratic Functions

The applications of quadratic functions extend far beyond the classroom and into the real world. These functions, with their distinctive parabolic curves, pop up in surprising places, from the trajectory of a ball to the design of bridges. Understanding quadratic functions isn't just an academic exercise; it's a key to unlocking a deeper understanding of the world around us.

One of the most classic examples is projectile motion. Imagine throwing a ball into the air. Its path follows a curve, and that curve can be accurately modeled by a quadratic function. The height of the ball at any given time can be predicted using a quadratic equation, taking into account factors like initial velocity and gravity. This principle is used in sports, engineering, and even military applications, like calculating the trajectory of a missile.

Quadratic functions also play a crucial role in engineering and architecture. Parabolic shapes are incredibly strong and efficient, which is why they're used in the design of bridges, arches, and satellite dishes. The shape of a suspension bridge cable, for example, closely resembles a parabola, distributing weight evenly and maximizing the bridge's stability. Similarly, the curved surface of a satellite dish is designed using a quadratic function to focus incoming signals onto a single point.

In the business world, quadratic functions are used to model profit and cost curves. A company might use a quadratic equation to determine the optimal price for a product, the price that will maximize their profit. The relationship between cost, revenue, and profit often follows a quadratic pattern, allowing businesses to make informed decisions about pricing and production.

Even in the seemingly unrelated field of computer graphics, quadratic functions play a role. They're used to create smooth curves and surfaces in 3D modeling and animation. By stringing together parabolic segments, artists and designers can create complex and visually appealing shapes.

The widespread applications of quadratic functions highlight their versatility and importance. From predicting the path of a projectile to designing a bridge, these functions are powerful tools for solving real-world problems.

In conclusion, by carefully calculating the missing values, completing the table, and understanding the key features of the graph, we've gained a solid understanding of the quadratic function f(x) = x^2 - 9. Keep practicing, and you'll be a quadratic function pro in no time!