Affine Function Analysis: Finding F(-2) With Decreasing Graph

Hey guys! Let's dive into a cool math problem involving affine functions! We're gonna break down how to find the value of f(-2) given some specific characteristics of the function. This is a classic example that combines understanding function behavior with some simple algebra. So, buckle up, and let's get started. We'll explore the key concepts, step-by-step solutions, and even a little bit of the function's graphical representation. This is how we are going to solve the problem for a better understanding.

The Core Concepts of Affine Functions and Linear Graphs

First off, what even is an affine function? An affine function is essentially a linear function. More precisely, an affine function, often written as f(x) = Ax + B, where A and B are constants, represents a straight line when graphed. The key here is that the 'A' determines the slope of the line, and 'B' tells us where the line crosses the y-axis (the y-intercept). If the slope (A) is negative, the line goes downwards as you move from left to right - we call this a decreasing function. That's a crucial piece of info for our problem, as we know the graph decreases. The root of the function is the x-value where f(x) equals zero – where the line crosses the x-axis. Thinking visually, we have a line that's going down, hitting the y-axis at a specific point, and also hitting the x-axis at a different point. That's the essence of what we're working with. Understanding these concepts is fundamental to solving the problem. The behavior of these functions is directly linked to their formulas and their graphs. The coefficient 'A' is extremely important, it tells us the inclination of the graph. The coefficient 'B' is also important, as it gives us the intersection of the function with the Y axis, and the root of the function is also a very important element to calculate the value of the function.

Now, let's look at the specific details the question gave us. We are told the function has a decreasing graph, which means 'A' must be negative. The graph intercepts the y-axis at the point (0, 4). This immediately gives us a vital piece of information – the y-intercept, which is the value of 'B'. Recall that the y-intercept is where the graph crosses the y-axis, and this occurs when x = 0. Therefore, when x = 0, f(x) = 4. This means that 'B' in our equation f(x) = Ax + B is equal to 4. Also, we are told that the root of the function is x = 2. A function's root is where f(x) = 0. So, when x = 2, f(x) = 0. These two key values, combined with our knowledge about the function type, make it possible to work out the function's formula and calculate f(-2).

Step-by-Step Solution: Unraveling the Affine Function

Alright, let's get down to business and find the value of f(-2). We've got our toolbox full of clues: we know the function is decreasing, the y-intercept is (0, 4), and the root is x = 2. Let's use this to solve the problem step by step. We'll build up our function equation, bit by bit. From the information provided, we know that the general form of our function is f(x) = Ax + B. We also know that the graph intercepts the y-axis at (0,4). This allows us to determine the value of 'B' in the equation, that is, when x=0, f(x) = 4. Since the y-intercept is (0,4), we know that B = 4. This simplifies our equation to f(x) = Ax + 4.

Next, the root of the function is where f(x) = 0. We're given that the root is x = 2. This means that when x = 2, f(x) = 0. Now we can substitute these values into our equation. So, substituting x = 2 and f(x) = 0 into f(x) = Ax + 4, we get 0 = 2A + 4. Now, we just need to solve this simple equation for 'A'.

Let's isolate 'A'. Subtract 4 from both sides: -4 = 2A. Then divide both sides by 2: A = -2. So, we've found our 'A' and we know that A = -2, which aligns perfectly with our initial thought that 'A' should be negative because the graph is decreasing. Now, we can rewrite our full equation with both 'A' and 'B' values in place. Our function is f(x) = -2x + 4. Now that we have the full equation, we can calculate the value of f(-2) by substituting x = -2 into our equation. So, f(-2) = -2*(-2) + 4. This results in f(-2) = 4 + 4, which means f(-2) = 8. Boom! We've found the solution.

Visualizing and Understanding the Results

It's always great to visualize your results. The function f(x) = -2x + 4 is a straight line. It has a y-intercept at (0, 4) – the point where the line crosses the y-axis. The line has a negative slope, meaning that it goes downwards as you move from left to right. The root of the function is x = 2, which corresponds to the point (2, 0) – where the line crosses the x-axis. We've determined that f(-2) = 8. This means that at the point x = -2, the y-value of our function is 8. You can also imagine this graphically as the point (-2, 8) on your graph of the function. This point lies on the line defined by the function. These points help you visualize the values and how they are used within the function itself.

To solidify your understanding, try plotting this function using an online graphing tool. You'll see the line decreasing, intercepting the y-axis at 4, and crossing the x-axis at 2. If you find plotting easy, you can also substitute other values for x to check your calculations. For example, f(1) = -2*1 + 4 = 2, which you can see at the point (1,2) on your graph. This confirms your calculated answers.

Conclusion

Great job, guys! We have successfully found the value of f(-2) for our affine function. This problem is a classic application of understanding linear functions, y-intercepts, and roots. Always remember that the information given to you, such as the y-intercept or the root, can be directly used to deduce the missing variables in an equation. Also, remember to take your time, organize your thoughts, and break down complex problems into manageable steps. If you take these steps when you are tackling more math problems, you should be able to solve them with ease. Keep practicing, keep exploring, and keep the mathematical spirit alive. You've got this!