Evaluating F(x) = 3x² + X - 3 For Various Inputs
In the fascinating world of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. Among these functions, quadratic functions hold a special place due to their unique properties and wide range of applications. Quadratic functions are defined by a polynomial equation of degree two, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a is not equal to zero. These functions exhibit a characteristic parabolic shape when graphed, making them crucial in fields like physics, engineering, and economics. This article will delve into the practical evaluation of a specific quadratic function, f(x) = 3x² + x - 3, for various input values. Understanding how to evaluate functions is essential for analyzing their behavior, solving equations, and applying them to real-world scenarios. So, let's embark on this mathematical journey and unravel the intricacies of evaluating quadratic functions.
a. Finding f(2)
To begin our exploration, let's evaluate the function f(x) at x = 2. This means we'll substitute the value 2 for every instance of x in the function's equation. Evaluating f(2) is a straightforward process that allows us to determine the output of the function for a specific input. So, let's dive into the calculations and uncover the value of f(2). We start with the given function:
f(x) = 3x² + x - 3
Now, substitute x with 2:
f(2) = 3(2)² + 2 - 3
Next, we perform the arithmetic operations following the order of operations (PEMDAS/BODMAS). First, we calculate the exponent:
f(2) = 3(4) + 2 - 3
Then, we perform the multiplication:
f(2) = 12 + 2 - 3
Finally, we perform the addition and subtraction from left to right:
f(2) = 14 - 3
f(2) = 11
Therefore, the value of the function f(x) = 3x² + x - 3 at x = 2 is 11. This means that when we input 2 into the function, the output is 11. Evaluating f(2) demonstrates the fundamental process of substituting a value into a function and simplifying the expression to obtain the corresponding output. This process is crucial for understanding the behavior of functions and their applications in various fields.
b. Determining f(0)
Next, let's investigate the value of the function f(x) when x = 0. This is a particularly interesting case as it often reveals important information about the function's behavior near the origin. Evaluating f(0) can help us identify the y-intercept of the function's graph, which is the point where the graph intersects the y-axis. So, let's substitute 0 for x in the function's equation and see what we get. Starting with the function:
f(x) = 3x² + x - 3
Substitute x with 0:
f(0) = 3(0)² + 0 - 3
Now, simplify the expression. The term 3(0)² becomes 0 since any number multiplied by 0 is 0:
f(0) = 0 + 0 - 3
Finally, perform the subtraction:
f(0) = -3
Thus, the value of the function f(x) = 3x² + x - 3 at x = 0 is -3. This tells us that the graph of the function intersects the y-axis at the point (0, -3). Evaluating f(0) is a simple yet powerful technique for finding the y-intercept of a function, which is a key feature for sketching the graph and understanding its overall behavior. In this case, we've found that when the input is 0, the output of the function is -3.
c. Calculating f(-4)
Now, let's shift our focus to evaluating f(x) at a negative value, specifically x = -4. This will provide us with further insights into the function's behavior and how it responds to negative inputs. Calculating f(-4) involves substituting -4 for every instance of x in the function's equation and then simplifying the resulting expression. So, let's embark on this calculation and discover the value of f(-4). We begin with the given function:
f(x) = 3x² + x - 3
Substitute x with -4:
f(-4) = 3(-4)² + (-4) - 3
Remember to pay close attention to the signs when dealing with negative numbers. First, we calculate the exponent. Squaring a negative number results in a positive number:
f(-4) = 3(16) + (-4) - 3
Next, perform the multiplication:
f(-4) = 48 - 4 - 3
Finally, perform the subtraction from left to right:
f(-4) = 44 - 3
f(-4) = 41
Therefore, the value of the function f(x) = 3x² + x - 3 at x = -4 is 41. This indicates that when we input -4 into the function, the output is 41. Calculating f(-4) demonstrates how to handle negative inputs when evaluating functions, emphasizing the importance of careful attention to signs and the order of operations. This calculation provides another data point for understanding the behavior of the function across different input values.
In conclusion, we have successfully evaluated the quadratic function f(x) = 3x² + x - 3 for three different input values: x = 2, x = 0, and x = -4. We found that f(2) = 11, f(0) = -3, and f(-4) = 41. These calculations demonstrate the fundamental process of substituting values into a function and simplifying the expression to obtain the corresponding output. Evaluating functions is a crucial skill in mathematics, allowing us to analyze their behavior, solve equations, and apply them to real-world scenarios. By understanding how a function responds to different inputs, we gain valuable insights into its properties and characteristics. This exercise has not only provided specific values for the function but also reinforced the importance of careful calculation and attention to detail when working with mathematical expressions. So, keep practicing and exploring the fascinating world of functions, and you'll be amazed at the power and versatility they offer!
