Analyzing The Quadratic Function F(x) = -3x² - 6x + 144 Direction Intercepts And Vertex
Hey guys! Let's dive into the fascinating world of quadratic functions with a detailed exploration of the function f(x) = -3x² - 6x + 144. This function, with its distinctive curve, holds a wealth of information that we can unravel. We'll examine its direction, pinpoint its intercepts, and locate its vertex. So, buckle up and let's embark on this mathematical journey together!
Understanding the Direction of the Graph
When we talk about the direction of a quadratic graph, we're essentially discussing whether the parabola opens upwards or downwards. This key characteristic is determined by the coefficient of the x² term in the quadratic equation. In our case, the function is f(x) = -3x² - 6x + 144. Notice that the coefficient of the x² term is -3. This negative value is a crucial indicator.
Direction of the Graph Explained: A negative coefficient for the x² term means the parabola opens downwards. Think of it like a frown – a negative leading coefficient makes the parabola "frown." Conversely, if the coefficient were positive, the parabola would open upwards, forming a smile. This simple rule helps us visualize the basic shape of the quadratic function's graph even before we plot any points. So, in the case of f(x) = -3x² - 6x + 144, the graph opens downwards, creating a maximum point rather than a minimum.
This downward-opening characteristic has significant implications. It tells us that the function has a maximum value, which occurs at the vertex of the parabola. This maximum value is the highest point the function reaches. In practical applications, this could represent the maximum height of a projectile, the peak profit for a business, or any other scenario where finding a maximum is crucial. Understanding the direction of the graph is the first step in unlocking the secrets hidden within the quadratic function.
The direction of the parabola isn't just a visual detail; it's a fundamental property that dictates the function's behavior. It influences whether the function has a maximum or minimum value, and it affects how the function increases or decreases as x changes. By simply observing the sign of the leading coefficient, we gain a powerful insight into the overall nature of the quadratic function. This is why it's so important to start our analysis by determining the direction of the graph – it sets the stage for everything else we'll discover.
Finding the Y-intercept
The y-intercept is the point where the graph of the function intersects the y-axis. It's a crucial point because it represents the value of the function when x is equal to 0. To find the y-intercept, we simply substitute x = 0 into our function f(x) = -3x² - 6x + 144.
Calculating the Y-intercept: Let's plug in x = 0:
f(0) = -3(0)² - 6(0) + 144
Simplifying this, we get:
f(0) = 0 - 0 + 144
f(0) = 144
Therefore, the y-intercept is at y = 144. This means the graph of the function crosses the y-axis at the point (0, 144). The y-intercept gives us a starting point for visualizing the graph of the quadratic function. It tells us where the parabola begins its journey on the coordinate plane. In many real-world scenarios, the y-intercept holds practical significance. For example, if this function represented the profit of a business, the y-intercept would indicate the initial profit (or loss) when no items have been sold.
The y-intercept is a relatively easy point to calculate, but its importance shouldn't be underestimated. It provides a direct connection between the algebraic representation of the function and its graphical representation. It's a fixed point that helps us anchor the parabola and understand its position on the coordinate plane. Furthermore, the y-intercept can be used in conjunction with other key features, such as the vertex and x-intercepts, to sketch an accurate graph of the quadratic function.
In the context of problem-solving, the y-intercept often provides a crucial piece of information. It can serve as a known value that helps us determine other unknowns, or it can be used to verify the accuracy of our calculations. For instance, if we were given a quadratic function and asked to find a point on the graph, calculating the y-intercept would be a logical first step. So, remember, the y-intercept is more than just a point on the graph – it's a valuable tool for understanding and working with quadratic functions.
Determining the X-intercepts
The x-intercepts, also known as the roots or zeros of the function, are the points where the graph intersects the x-axis. At these points, the value of the function, f(x), is equal to 0. Finding the x-intercepts is a bit more involved than finding the y-intercept, but it's a crucial step in fully understanding the behavior of the quadratic function.
Methods for Finding X-intercepts: To find the x-intercepts, we need to solve the equation f(x) = 0. This means we need to find the values of x that make the equation -3x² - 6x + 144 = 0 true. There are several methods we can use to solve quadratic equations, including:
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Factoring: This method involves rewriting the quadratic expression as a product of two linear expressions. If we can factor the quadratic, we can set each factor equal to zero and solve for x.
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Quadratic Formula: This formula provides a direct solution for x in any quadratic equation of the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / (2a)
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Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial. While it's a bit more complex, it can be useful in certain situations.
Applying the Quadratic Formula: In our case, the quadratic equation is -3x² - 6x + 144 = 0. Let's use the quadratic formula to find the x-intercepts. Here, a = -3, b = -6, and c = 144. Plugging these values into the formula, we get:
x = (6 ± √((-6)² - 4(-3)(144))) / (2(-3))
Simplifying this expression:
x = (6 ± √(36 + 1728)) / (-6)
x = (6 ± √1764) / (-6)
x = (6 ± 42) / (-6)
Now we have two possible solutions:
x₁ = (6 + 42) / (-6) = 48 / (-6) = -8
x₂ = (6 - 42) / (-6) = -36 / (-6) = 6
Therefore, the x-intercepts are at x = -8 and x = 6. These are the points where the parabola crosses the x-axis. The x-intercepts provide valuable information about the function's behavior. They tell us where the function's value is zero, which can be significant in various applications. For example, if this function represented the height of a projectile, the x-intercepts would indicate when the projectile hits the ground.
Understanding the x-intercepts in conjunction with the y-intercept and the vertex gives us a comprehensive picture of the quadratic function's graph. These key points allow us to sketch the parabola accurately and interpret its meaning in a given context. So, mastering the techniques for finding x-intercepts is essential for anyone working with quadratic functions.
Locating the Vertex
The vertex is the most crucial point on the parabola. It represents either the maximum or minimum value of the quadratic function. Since we know our parabola opens downwards (because the coefficient of x² is negative), the vertex will be the highest point on the graph, representing the maximum value of the function. The vertex is essentially the turning point of the parabola – the point where the function changes direction.
Finding the Vertex: The vertex of a parabola can be found using a specific formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex, often denoted as h, is given by:
h = -b / (2a)
Once we find the x-coordinate, we can substitute it back into the original function to find the y-coordinate of the vertex, often denoted as k. So, the vertex is the point (h, k).
Calculating the Vertex for Our Function: Let's apply this to our function, f(x) = -3x² - 6x + 144. Here, a = -3 and b = -6. So, the x-coordinate of the vertex is:
h = -(-6) / (2(-3))
h = 6 / (-6)
h = -1
Now, let's find the y-coordinate by substituting x = -1 into the function:
k = f(-1) = -3(-1)² - 6(-1) + 144
k = -3(1) + 6 + 144
k = -3 + 6 + 144
k = 147
Therefore, the vertex is at the point (-1, 147). This point represents the maximum value of the function. The vertex is not just a point on the graph; it's a key feature that reveals the function's extreme value. In many real-world applications, finding the vertex is the primary goal. For example, if the function represented the profit of a business, the vertex would indicate the point where the maximum profit is achieved. Similarly, if the function represented the height of a projectile, the vertex would indicate the maximum height reached.
The vertex, in conjunction with the x-intercepts and the y-intercept, provides a complete understanding of the quadratic function's behavior. It allows us to sketch the parabola accurately and interpret its meaning in a given context. The vertex also plays a crucial role in understanding the symmetry of the parabola. The vertical line that passes through the vertex is the axis of symmetry, which divides the parabola into two mirror images. So, mastering the techniques for finding the vertex is essential for anyone working with quadratic functions and their applications.
By finding the vertex, we gain a deeper understanding of the function's overall behavior and its maximum or minimum value. This information is invaluable in a wide range of applications, from physics and engineering to economics and finance.
Conclusion
We've successfully explored the quadratic function f(x) = -3x² - 6x + 144, uncovering its key features. We determined that the graph opens downwards, found the y-intercept at y = 144, calculated the x-intercepts at x = -8 and x = 6, and located the vertex at (-1, 147). This comprehensive analysis provides a complete picture of the function's behavior and its graphical representation. Understanding these elements allows us to effectively work with quadratic functions in various mathematical and real-world contexts. Keep exploring, guys, and you'll master these concepts in no time!
