Finding A Linear Function: F(-5)=2 & F(5)=5
Hey guys! Let's dive into a classic math problem: finding the equation of a linear function. This is something you'll encounter a lot in algebra and beyond, so it's super important to get a solid handle on it. In this case, we're given two points on the line, f(-5) = 2 and f(5) = 5, and our mission is to determine the equation for f(x). Think of it like connecting the dots – we have two dots, and we need to draw the straight line that goes through them and then express that line as an equation. Sounds fun, right? This might seem tricky at first, but I promise it's totally doable once we break it down into simple steps. We'll be using the slope-intercept form of a line, which is a friendly way to represent linear equations. So, grab your pencils, and let's get started on unraveling this linear function mystery!
Understanding Linear Functions
So, what exactly is a linear function? It's basically a function that, when graphed, forms a straight line. Think of a perfectly straight road stretching out into the distance – that's the visual representation of a linear function. The general form of a linear function is f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept. The slope tells us how steep the line is, and the y-intercept is the point where the line crosses the vertical y-axis. Understanding this form is key to solving our problem. We need to figure out the values of 'm' and 'b' to define our specific linear function. Now, why is this important? Well, linear functions are everywhere! They model real-world scenarios like the constant speed of a car, the steady growth of a plant, or even the simple relationship between hours worked and money earned. Mastering linear functions opens the door to understanding and predicting many patterns around us. So, let's get comfortable with 'm' and 'b' – they're our trusty guides in the world of linear equations. We'll use the given information, f(-5) = 2 and f(5) = 5, as coordinates to pinpoint our line's unique characteristics. The beauty of linear functions lies in their simplicity and predictability, making them a foundational concept in mathematics and its applications.
Step 1: Calculate the Slope (m)
The slope, often represented by the letter 'm', is the heart of a linear function. It tells us how much the function's value changes (the rise) for every unit change in the input (the run). In simpler terms, it's the steepness of the line. To calculate the slope, we use the formula: m = (y2 - y1) / (x2 - x1). This might look a bit intimidating, but it's actually quite straightforward. We're given two points, f(-5) = 2 and f(5) = 5, which we can rewrite as coordinates: (-5, 2) and (5, 5). Think of these as (x1, y1) and (x2, y2). Now, let's plug these values into our slope formula: m = (5 - 2) / (5 - (-5)) = 3 / 10. So, our slope, 'm', is 3/10. This means that for every 10 units we move horizontally along the x-axis, the line rises 3 units vertically along the y-axis. Now that we've found the slope, we're one step closer to cracking the code of our linear function. Remember, the slope is constant throughout the entire line, so this value will be crucial in determining the full equation. Next up, we'll use this slope along with one of our points to find the y-intercept.
Step 2: Find the Y-Intercept (b)
The y-intercept, denoted by 'b', is the point where our line crosses the y-axis. It's the value of f(x) when x is 0. Finding the y-intercept is like finding the starting point of our line on the graph. To find 'b', we can use the slope-intercept form of the equation, f(x) = mx + b, and plug in the slope we just calculated (m = 3/10) and one of our given points. Let's use the point (5, 5). So, we have: 5 = (3/10) * 5 + b. Now, we solve for 'b': 5 = 3/2 + b. To isolate 'b', we subtract 3/2 from both sides: b = 5 - 3/2. Converting 5 to a fraction with a denominator of 2, we get: b = 10/2 - 3/2 = 7/2. Therefore, our y-intercept, 'b', is 7/2. This means the line crosses the y-axis at the point (0, 7/2). Now we have both the slope (m = 3/10) and the y-intercept (b = 7/2). We're in the home stretch! With these two pieces of information, we can construct the full equation of our linear function. The y-intercept gives us the vertical positioning of the line, complementing the slope's directional information.
Step 3: Write the Equation for f(x)
Alright, guys, this is the moment we've been working towards! We've found the slope (m = 3/10) and the y-intercept (b = 7/2). Now, we can finally write the equation for our linear function, f(x). Remember the slope-intercept form: f(x) = mx + b. We simply plug in the values we've calculated for 'm' and 'b'. So, here it is: f(x) = (3/10)x + 7/2. Ta-da! We've successfully found the equation that represents the line passing through the points (-5, 2) and (5, 5). This equation tells us everything we need to know about this specific linear function. For any value of x, we can plug it into this equation and find the corresponding value of f(x). It's like having a roadmap for the line! This equation is the final answer to our problem. It neatly encapsulates the relationship between x and f(x) for this particular linear function. We started with two points and ended up with a complete equation – pretty cool, huh? The ability to find such equations is a powerful tool in mathematics and its applications.
Verification
Before we celebrate our victory, let's do a quick verification to make sure our equation is spot-on. It's always a good idea to double-check your work, especially in math! We can do this by plugging our original x-values, -5 and 5, back into our equation, f(x) = (3/10)x + 7/2, and see if we get the corresponding f(x) values, 2 and 5, respectively. Let's start with x = -5: f(-5) = (3/10) * (-5) + 7/2 = -3/2 + 7/2 = 4/2 = 2. Great! It checks out. Now, let's try x = 5: f(5) = (3/10) * 5 + 7/2 = 3/2 + 7/2 = 10/2 = 5. Awesome! It checks out again. Since our equation produces the correct f(x) values for both given x values, we can be confident that we've found the correct linear function. This step is crucial because it confirms that our calculations are accurate and our equation truly represents the line defined by the given points. Verification is a valuable practice in problem-solving, ensuring the reliability of our results. So, with this double-check, we can confidently say we've nailed it!
Conclusion
So, there you have it, guys! We successfully navigated the world of linear functions and found the equation for f(x) given f(-5) = 2 and f(5) = 5. We started by understanding what linear functions are, then calculated the slope, found the y-intercept, wrote the equation, and even verified our answer. That's a full journey through linear function territory! Remember, the key takeaways here are the slope-intercept form (f(x) = mx + b) and the steps involved in finding 'm' and 'b'. These skills will serve you well in many mathematical adventures to come. Linear functions are not just abstract concepts; they're tools for understanding and modeling real-world relationships. By mastering these fundamentals, you're building a strong foundation for more advanced mathematical topics. So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this! If you get stuck on a similar problem, just remember the steps we've covered, and you'll be able to crack the code of any linear function that comes your way. Keep up the awesome work!