Analyzing Y = -3x^2 + 36x - 93: A Quadratic Deep Dive

by Alex Johnson 54 views

Let's dive deep into the world of quadratic equations! In this article, we're going to explore the equation y = -3x^2 + 36x - 93. We'll break it down, discuss its key features, and understand how to analyze it effectively. Whether you're a student tackling algebra or just someone curious about math, this guide is for you. So, grab your thinking cap, and let's get started!

Understanding the Basics of Quadratic Equations

First off, let's establish what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is y = ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, it would turn into a linear equation, which is a whole different ball game.

In our equation, y = -3x^2 + 36x - 93, we can easily identify the constants: a = -3, b = 36, and c = -93. These constants play a crucial role in determining the shape and position of the parabola, which is the graph of a quadratic equation. The coefficient 'a' is particularly important because it tells us whether the parabola opens upwards (if a > 0) or downwards (if a < 0). Since our 'a' is -3, which is less than zero, we know our parabola will open downwards, meaning it has a maximum point.

Understanding these fundamental concepts is vital before we proceed further. Quadratic equations pop up everywhere in real-world applications, from physics (projectiles, trajectories) to engineering (designing curves) and even economics (modeling costs and revenues). So, having a solid grasp of the basics will serve you well. The coefficients also greatly influence the vertex and axis of symmetry, which we'll touch on later. It's not just about the numbers; it's about how they interact to shape the graph and, consequently, the solutions of the equation. Remember, a negative 'a' flips the parabola, changing its entire outlook. This initial understanding is the bedrock for more advanced analysis, and it's where our journey truly begins.

Key Features of y = -3x^2 + 36x - 93

Now, let’s zoom in on our specific equation: y = -3x^2 + 36x - 93. What are the key features we should be looking at? There are several aspects that help us understand this equation better. These features provide a comprehensive picture of the equation’s behavior and its graphical representation.

1. The Vertex

The vertex is the most crucial point of a parabola. It represents the maximum or minimum value of the quadratic function. In our case, since the parabola opens downwards (a = -3), the vertex will be the maximum point. The vertex is essentially the peak of our curve. To find the vertex, we first need to find the x-coordinate, which can be calculated using the formula x = -b / 2a. Plugging in our values, we get x = -36 / (2 * -3) = -36 / -6 = 6. So, the x-coordinate of the vertex is 6.

To find the y-coordinate, we substitute this x-value back into our equation: y = -3(6)^2 + 36(6) - 93. This simplifies to y = -3(36) + 216 - 93 = -108 + 216 - 93 = 15. Therefore, the vertex of our parabola is at the point (6, 15). This point is the highest point on our graph, and it gives us a clear idea of the function's maximum output.

Understanding the vertex is incredibly useful. It not only tells us the maximum value of the function but also serves as a reference point for sketching the graph. The vertex is where the parabola changes direction, making it a critical feature for analysis and application. Its location is a direct reflection of the 'a' and 'b' coefficients, highlighting how these values shape the parabola's characteristics.

2. The Axis of Symmetry

The axis of symmetry is an imaginary vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. It's like a mirror line; whatever is on one side of the parabola is mirrored on the other side. The equation of the axis of symmetry is simply x = the x-coordinate of the vertex. In our case, the axis of symmetry is x = 6. This line is incredibly helpful because it provides a visual guide to the parabola’s symmetry and helps in plotting points.

The axis of symmetry is not just a theoretical concept; it has practical implications. For instance, if you know a point on one side of the parabola, you automatically know its corresponding point on the other side, thanks to this symmetry. This can significantly reduce the amount of calculation needed when graphing the function. The axis of symmetry reinforces the idea that parabolas are beautifully balanced curves, and understanding this symmetry can greatly simplify problem-solving.

3. The Y-intercept

The y-intercept is the point where the parabola intersects the y-axis. It's the value of y when x = 0. To find the y-intercept, we simply substitute x = 0 into our equation: y = -3(0)^2 + 36(0) - 93. This simplifies to y = -93. So, the y-intercept is the point (0, -93). The y-intercept gives us an initial point on the graph and provides a sense of the function's behavior near the y-axis.

The y-intercept is particularly useful because it’s straightforward to calculate. It also gives us a concrete point to start with when sketching the parabola. This point, along with the vertex, starts to paint a picture of the parabola’s position and orientation on the coordinate plane. It's a simple but powerful piece of information that helps ground our understanding of the function's graph.

4. The X-intercepts (Roots or Zeros)

The x-intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis. These are the values of x when y = 0. To find the x-intercepts, we need to solve the quadratic equation 0 = -3x^2 + 36x - 93. This can be done using several methods, such as factoring, completing the square, or using the quadratic formula. The quadratic formula is particularly useful and is given by:

x = [-b ± √(b^2 - 4ac)] / 2a

Plugging in our values (a = -3, b = 36, c = -93), we get:

x = [-36 ± √(36^2 - 4(-3)(-93))] / (2 * -3)

x = [-36 ± √(1296 - 1116)] / -6

x = [-36 ± √180] / -6

x = [-36 ± 6√5] / -6

Simplifying this, we get two x-intercepts:

x = 6 - √5 and x = 6 + √5

So, the x-intercepts are approximately (3.76, 0) and (8.24, 0). These points are where the parabola crosses the x-axis, and they are crucial for understanding the solutions of the equation. They tell us the values of x for which the function's output is zero.

The x-intercepts provide critical insights into the quadratic equation's solutions. They are the roots of the equation, and their presence or absence can tell us a lot about the nature of the parabola. If the discriminant (the part under the square root in the quadratic formula, b^2 - 4ac) is positive, there are two distinct x-intercepts. If it's zero, there is one x-intercept (the vertex touches the x-axis), and if it's negative, there are no real x-intercepts (the parabola doesn't cross the x-axis). In our case, since we have two real x-intercepts, we know the parabola crosses the x-axis at two points, providing a complete picture of the function’s behavior.

Analyzing the Graph of y = -3x^2 + 36x - 93

Now that we've identified the key features, let's talk about what the graph of y = -3x^2 + 36x - 93 looks like. We know it's a parabola that opens downwards, with a vertex at (6, 15), an axis of symmetry at x = 6, a y-intercept at (0, -93), and x-intercepts at approximately (3.76, 0) and (8.24, 0). Putting all this information together, we can sketch the graph.

Sketching the Graph

  1. Plot the Vertex: Start by plotting the vertex at (6, 15). This is the highest point on the graph.
  2. Draw the Axis of Symmetry: Draw a vertical dashed line through x = 6. This line divides the parabola into two symmetrical halves.
  3. Plot the Y-intercept: Plot the y-intercept at (0, -93). This point is quite far down the y-axis, indicating the parabola extends quite low.
  4. Plot the X-intercepts: Plot the x-intercepts at approximately (3.76, 0) and (8.24, 0).
  5. Sketch the Parabola: Now, draw a smooth curve connecting these points, making sure the parabola opens downwards and is symmetrical about the axis of symmetry.

The graph will show a downward-facing parabola with its peak at the vertex (6, 15). It will cross the x-axis at the two x-intercepts and intersect the y-axis far below the x-axis. The steepness of the curve is influenced by the coefficient 'a' (-3), which makes the parabola a bit narrower compared to a parabola with a smaller absolute value of 'a'.

Interpreting the Graph

The graph visually represents the behavior of the function. The vertex (6, 15) indicates the maximum value of the function is 15, which occurs when x = 6. The x-intercepts show where the function's value is zero, which are the solutions to the equation 0 = -3x^2 + 36x - 93. The y-intercept shows the value of the function when x is zero. The symmetry of the graph around the axis of symmetry reinforces the balanced nature of quadratic functions.

By analyzing the graph, we gain a deeper understanding of the function's characteristics and behavior. The visual representation complements the algebraic analysis, making it easier to grasp the concepts and their implications. The graph not only provides a visual confirmation of our calculations but also offers intuitive insights into the function's properties.

Real-World Applications

Quadratic equations aren't just abstract mathematical concepts; they have numerous real-world applications. Understanding them can help us solve practical problems in various fields. Let's explore a few examples.

1. Projectile Motion

One of the most common applications of quadratic equations is in physics, specifically in projectile motion. The path of a projectile (like a ball thrown in the air) can be modeled by a parabola. The equation y = -3x^2 + 36x - 93 could, for instance, represent the trajectory of an object, where y is the height and x is the horizontal distance. The vertex would represent the maximum height the object reaches, and the x-intercepts would represent the points where the object hits the ground.

In this context, understanding the vertex is crucial. It helps determine the highest point the projectile will reach, which is often a critical piece of information in planning and executing various tasks. The x-intercepts tell us the range of the projectile, and the equation itself can be used to calculate the height at any given horizontal distance. Projectile motion is a prime example of how abstract mathematical concepts can be applied to predict and analyze real-world phenomena.

2. Optimization Problems

Quadratic equations are also used in optimization problems, where the goal is to find the maximum or minimum value of a function. Businesses, for example, might use quadratic equations to model costs, revenues, or profits. The vertex of the parabola would then represent the optimal value—the maximum profit or the minimum cost. Our equation y = -3x^2 + 36x - 93 could represent a profit function, where y is the profit and x is the number of units sold. The vertex (6, 15) would indicate that the maximum profit is 15, which is achieved when 6 units are sold.

Optimization is at the heart of many business and engineering decisions. Knowing the maximum or minimum value of a function can drive strategic planning and resource allocation. Quadratic equations provide a powerful tool for modeling and solving these types of problems, making them invaluable in decision-making processes. This application highlights the practical relevance of understanding quadratic functions in real-world scenarios.

3. Engineering and Design

Engineers and designers often use quadratic equations to design curves and shapes. Bridges, arches, and even satellite dishes are often designed using parabolic shapes. The properties of parabolas, such as their symmetry and focus, make them ideal for these applications. For instance, the reflective surface of a satellite dish is a paraboloid, which is a 3D version of a parabola. The focus of the parabola is where the signal is concentrated, making it highly efficient for receiving and transmitting signals.

In engineering and design, the precision offered by quadratic equations is essential. The ability to model and predict curves accurately is crucial for ensuring structural integrity and functional efficiency. From the elegant curves of bridges to the precise shapes of antennas, quadratic equations play a fundamental role in shaping the world around us. This application underscores the critical role of mathematical principles in practical design and engineering.

Conclusion

Analyzing the quadratic equation y = -3x^2 + 36x - 93 involves understanding its key features: the vertex, axis of symmetry, y-intercept, and x-intercepts. By finding these elements, we can sketch the graph and interpret its behavior. Quadratic equations have numerous real-world applications, from modeling projectile motion to solving optimization problems and designing engineering structures. Mastering quadratic equations is not just an academic exercise; it's a valuable skill that can be applied in various fields.

Understanding the fundamentals of quadratic equations opens up a world of possibilities, both in mathematics and in practical applications. From the trajectory of a ball to the design of a bridge, quadratic functions play a pivotal role in our understanding and shaping of the world. So, keep exploring, keep questioning, and keep applying these concepts to see the beauty and power of mathematics in action. To further explore the world of quadratic equations and their applications, consider visiting trusted resources like Khan Academy's Quadratic Equations Section for more in-depth explanations and examples.