Evaluating F(x) = -2|x| + 8 At X = -1.8
Let's dive into evaluating the function f(x) = -2|x| + 8 at a specific point, x = -1.8. This involves understanding the absolute value function and how it affects the overall result. This article will provide a step-by-step guide on how to substitute the value of x into the function and simplify the expression to find the corresponding value of f(-1.8). Understanding how to evaluate functions is a fundamental skill in mathematics, and this example will help solidify your understanding of absolute value functions in particular. We will break down the process into manageable steps, ensuring clarity and comprehension for anyone looking to master this concept.
Understanding the Function
To begin, let's break down the function f(x) = -2|x| + 8. The core component here is the absolute value function, denoted by |x|. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. For example, |3| = 3 and |-3| = 3. In our function, the absolute value of x is multiplied by -2, which will reflect the absolute value across the x-axis and stretch it vertically by a factor of 2. Finally, we add 8 to the result, which shifts the entire function upward by 8 units on the y-axis.
The presence of the absolute value function gives f(x) a V-shape when graphed. The vertex of this V-shape will be at the point where x = 0, as that's where the absolute value part has the least effect. The coefficient -2 in front of the absolute value signifies that the V opens downwards, and the +8 part indicates that the entire graph is shifted 8 units up the y-axis. Visualizing the function in this way can help you anticipate the general behavior and range of its values. Understanding the transformations applied to the basic absolute value function—reflection, stretching, and shifting—is crucial for effectively evaluating it at different points.
When dealing with functions that include absolute values, it's important to carefully consider the sign of x. This is because the absolute value function behaves differently for positive and negative inputs. For positive values of x, |x| is simply x. However, for negative values of x, |x| becomes -x, which effectively changes the sign of the input. In our case, since we are evaluating f(-1.8), we need to remember that |-1.8| will be the positive value 1.8. This distinction is critical for arriving at the correct answer. Always take a moment to think about how the absolute value affects the expression, especially when dealing with negative values.
Substituting x = -1.8 into the Function
Now that we have a solid understanding of the function, the next step is to substitute x = -1.8 into the expression f(x) = -2|x| + 8. This means we replace every instance of x in the function with -1.8. When substituting, it's a good practice to use parentheses to avoid sign errors, especially when dealing with negative numbers. So, our expression becomes:
f(-1.8) = -2|-1.8| + 8
This substitution is the crucial first step in evaluating the function at a specific point. It translates the symbolic expression into a numerical calculation. The key here is to be meticulous and ensure that you replace every instance of x with the given value. Once you've made the substitution, the problem becomes a matter of simplifying the resulting expression using the order of operations. This careful substitution is a cornerstone of function evaluation and sets the stage for the subsequent arithmetic.
The next step after substitution is to evaluate the absolute value. Remember that the absolute value of a number is its distance from zero, so it's always non-negative. In our case, we need to find |-1.8|. Since -1.8 is 1.8 units away from zero, its absolute value is 1.8. So, |-1.8| = 1.8. Replacing this in our expression, we now have:
f(-1.8) = -2(1.8) + 8
This simplification is a direct application of the definition of absolute value. It's a critical step because it removes the absolute value bars, allowing us to proceed with standard arithmetic operations. Understanding how to correctly evaluate absolute values is essential for working with functions that involve them. By taking the time to clearly evaluate this part, you minimize the risk of errors in the subsequent calculations.
Simplifying the Expression
With the absolute value evaluated, our expression is now f(-1.8) = -2(1.8) + 8. Following the order of operations (PEMDAS/BODMAS), we need to perform the multiplication before the addition. So, we multiply -2 by 1.8:
-2 * 1.8 = -3.6
Now our expression looks like:
f(-1.8) = -3.6 + 8
Performing the multiplication first is crucial for getting the correct answer. It's a basic arithmetic principle, but it's important to adhere to it strictly. This multiplication step scales the absolute value by a factor of -2, and this result will then be combined with the constant term, 8. Getting this multiplication right is a key stepping stone to the final solution.
The final step is to perform the addition. We are adding -3.6 and 8. This is the same as subtracting 3.6 from 8:
8 - 3.6 = 4.4
Therefore,
f(-1.8) = 4.4
This addition brings everything together, combining the scaled absolute value with the constant term. It’s a straightforward arithmetic operation, but it's the culmination of all the previous steps. The result, 4.4, is the value of the function f(x) when x is -1.8. This is the final answer to our problem, and it represents a specific point on the graph of the function.
The Final Answer
After substituting x = -1.8 into the function f(x) = -2|x| + 8 and simplifying, we find that:
f(-1.8) = 4.4
This means that the value of the function at x = -1.8 is 4.4. In graphical terms, this corresponds to the point (-1.8, 4.4) on the graph of the function. Understanding how to evaluate functions at specific points is a fundamental skill in mathematics, and this example demonstrates the process clearly and concisely. We started by understanding the components of the function, particularly the absolute value, and then carefully substituted the given value of x. After simplifying step-by-step, we arrived at the solution. This approach can be applied to evaluating other functions as well, making it a valuable technique to master.
The result, 4.4, provides us with a specific point on the function's graph. This point gives us insight into the function's behavior around x = -1.8. Knowing the value of a function at different points allows us to sketch the graph, identify trends, and make predictions about the function's output for other inputs. In many real-world applications, functions model various phenomena, and evaluating them at specific points can provide valuable information about those phenomena. This exercise not only provides a numerical answer but also reinforces the connection between functions, their graphs, and their applications.
In summary, evaluating f(x) = -2|x| + 8 at x = -1.8 involves a series of steps: understanding the function, substituting the value, evaluating the absolute value, performing the multiplication, and finally, completing the addition. Each step builds upon the previous one, and careful attention to detail is crucial for arriving at the correct answer. This process highlights the importance of both algebraic manipulation and arithmetic precision in mathematics. By practicing similar problems, you can strengthen your understanding of function evaluation and become more confident in your mathematical abilities. For further information on functions and their evaluation, you can refer to resources like Khan Academy's Function Evaluation Section.
