Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically focusing on the equation y = x² - 4. This equation represents a parabola, a U-shaped curve that pops up all over the place in mathematics, physics, and even real-life scenarios like the trajectory of a ball thrown in the air. Our mission is to understand this function better by filling in a table of values. This involves finding the y-values that correspond to given x-values, and vice versa. It's like solving a puzzle where each piece fits perfectly to reveal the bigger picture of the function's behavior. We will learn how to find y when we know x, and also how to work backward to find x when y is given. This skill is super useful for graphing the function, understanding its properties, and even solving related equations. So, let's grab our mathematical tools and get started on this exciting journey of exploring y = x² - 4!
Understanding the Quadratic Function y = x² - 4
The quadratic function y = x² - 4 is a classic example of a parabola. The x² term is what gives it its curved shape, and the - 4 shifts the entire parabola down by four units on the y-axis. This shift is crucial because it determines where the vertex, the lowest point of the parabola, is located. In this case, the vertex is at (0, -4). Understanding this basic structure allows us to predict how the function will behave for different x-values. For instance, as x moves away from 0 in either the positive or negative direction, the x² term grows rapidly, causing y to increase. This symmetrical increase creates the characteristic U-shape. The axis of symmetry, which runs vertically through the vertex, is the line x = 0, meaning the parabola is mirrored across the y-axis. To truly master this function, we need to be comfortable plugging in values for x and solving for y, as well as working in reverse. This understanding of the graph's symmetry and the impact of the constant term will be invaluable as we fill in the missing values in our table. By visualizing the parabola, we can anticipate the general pattern of the values, making our calculations more intuitive and less prone to errors. Remember, mathematics is all about connecting the dots, and in this case, the dots are the (x, y) pairs that define our quadratic function.
Filling the Table: Finding y for Given x
Okay, let's get practical and start filling in our table. We're given a few x-values: -7, -5, and 0. Our mission is to find the corresponding y-values using the equation y = x² - 4. It's like a mathematical treasure hunt where each x-value is a clue leading us to its y-value counterpart. First up, let's tackle x = -7. We substitute -7 into our equation: y = (-7)² - 4. Remember that squaring a negative number results in a positive number, so (-7)² = 49. Then, we subtract 4: y = 49 - 4 = 45. So, when x = -7, y = 45. Next, let's try x = -5. We plug -5 into the equation: y = (-5)² - 4. Again, squaring -5 gives us 25, so y = 25 - 4 = 21. This means when x = -5, y = 21. Lastly, let's find the y-value for x = 0. This is a simple one: y = (0)² - 4 = -4. This result is significant because it tells us that the parabola's vertex, the lowest point, lies on the y-axis at y = -4. By performing these calculations, we're not just filling in blanks; we're building a deeper understanding of how the function behaves. Each point we find gives us a snapshot of the parabola's curve, and as we fill in more points, the picture becomes clearer. These calculations are the building blocks for graphing the function and making predictions about its behavior.
Filling the Table: Finding x for Given y
Now, let's switch gears and tackle the reverse problem: finding x-values when we're given y-values. This is like solving a mathematical mystery, where we have the answer but need to figure out the question. We have two y-values to work with: 21 and 96. Let's start with y = 21. We substitute this into our equation: 21 = x² - 4. Our goal is to isolate x. First, we add 4 to both sides of the equation: 21 + 4 = x², which simplifies to 25 = x². Now, we need to find the number that, when squared, equals 25. This is where the square root comes in. The square root of 25 is both 5 and -5, because both 5² and (-5)² equal 25. So, when y = 21, x can be either 5 or -5. This highlights an important property of parabolas: for a given y-value (except for the y-value at the vertex), there are usually two corresponding x-values due to the symmetry of the curve. Next, let's find the x-values for y = 96. We substitute 96 into the equation: 96 = x² - 4. Adding 4 to both sides gives us 100 = x². Taking the square root of both sides, we find that x can be either 10 or -10, since both 10² and (-10)² equal 100. So, when y = 96, x can be 10 or -10. This process of finding x for given y is a fundamental skill in algebra and calculus. It allows us to determine the points where the parabola intersects a horizontal line, which has applications in various fields, from engineering to economics.
Completed Table and What It Tells Us
Alright, let's gather all our findings and present the completed table. This table is like a map of our function, showing us exactly where it goes for specific x and y coordinates. Here's the filled-in table:
| x | -7 | -5 | 0 | 5 | 10 |
|---|---|---|---|---|---|
| y | 45 | 21 | -4 | 21 | 96 |
Looking at this table, we can see some key patterns. Notice the symmetry around x = 0. For instance, when x = -5 and x = 5, y is the same (21). This is a direct result of the parabolic shape. The vertex, the lowest point of the parabola, is clearly visible at (0, -4). As x moves away from 0 in either direction, y increases, confirming our earlier observations about the function's behavior. This table is not just a collection of numbers; it's a visual representation of the relationship between x and y defined by our quadratic function. We can use this table to plot the points on a graph and sketch the parabola. The more points we have, the more accurate our graph will be. The table also helps us understand the rate of change of the function. As x increases, y increases at an accelerating rate, which is characteristic of quadratic functions. By analyzing the table, we can make predictions about the function's behavior for other values of x. For example, we can reasonably guess that when x = -10, y will be 96, mirroring the result we found for x = 10. This ability to interpret and extrapolate from a table of values is a crucial skill in mathematics and data analysis.
Real-World Applications of Quadratic Functions
Guys, quadratic functions aren't just abstract mathematical concepts; they're all around us in the real world! You might be surprised at how often these parabolas pop up in everyday life. One of the most common examples is projectile motion. When you throw a ball, the path it follows through the air is a parabola. The quadratic function can help us model this path, predicting how far the ball will travel and how high it will go. Architects and engineers use quadratic functions to design arches and bridges. The parabolic shape is incredibly strong and efficient for distributing weight, making it ideal for these structures. Satellite dishes and radar antennas also utilize the parabolic shape to focus signals. The curved surface reflects incoming waves to a single point, maximizing the signal strength. In business and economics, quadratic functions can be used to model profit and cost curves. Understanding the vertex of the parabola, which represents the maximum or minimum value, can help businesses optimize their operations and maximize their profits. Even in the world of computer graphics and video games, quadratic functions play a crucial role in creating smooth curves and realistic trajectories. From the graceful arc of a fountain's spray to the trajectory of a rocket launch, quadratic functions provide a powerful tool for understanding and modeling the world around us. By mastering these functions, we gain a deeper appreciation for the mathematical principles that govern our physical reality. So, next time you see a parabola, remember that it's not just a curve; it's a mathematical story unfolding before your eyes.
So, guys, we've journeyed through the quadratic function y = x² - 4, filling in our table of values and uncovering the secrets of this elegant curve. We started by understanding the basic structure of the function, recognizing the importance of the x² term and the constant -4. We then put our skills to the test, finding y-values for given x-values and vice versa. This involved substituting values into the equation, solving for unknowns, and understanding the symmetry inherent in parabolas. Our completed table provided a visual representation of the function's behavior, highlighting the vertex, the axis of symmetry, and the increasing rate of change. But more importantly, we've seen that quadratic functions aren't just abstract concepts; they're powerful tools for modeling the real world. From projectile motion to architectural design, these parabolas are all around us, shaping the world we live in. By mastering the skills we've practiced today, you're not just learning math; you're gaining a new lens through which to view the world. Keep exploring, keep questioning, and keep applying your mathematical knowledge to the wonders around you. The world is full of mathematical puzzles waiting to be solved, and you now have the tools to tackle them!