Find The Formula Of A Linear Function

Let's dive into finding the formula for a linear function, guys! It sounds trickier than it is, trust me. We're given some clues, and it's our job to piece them together. So, let's break it down step by step.

Understanding Linear Functions

First, let's remember what a linear function actually is. A linear function is basically a straight line when you graph it. The general form looks like this: f(x) = mx + b, where m is the slope (or gradient) of the line, and b is the y-intercept (where the line crosses the y-axis).

The slope, m, tells us how steep the line is. A large positive m means the line goes up steeply as you move from left to right. A negative m means the line goes down. A zero m means it's a flat horizontal line. The y-intercept, b, is simply the value of f(x) when x is zero. It's the point (0, b) on the graph.

Our goal is to find these two values, m and b, using the information we've been given. We know that f(1) = 1 and f(2) = 5. That means when x is 1, the function's value is 1, and when x is 2, the function's value is 5. We can use these two points to figure out the slope and y-intercept.

Why is understanding linear functions important? Well, they show up everywhere in real life! From calculating the cost of something based on the quantity you buy, to predicting how far a car will travel at a constant speed, linear functions are super useful. They're a foundational concept in algebra, so mastering them is definitely worth the effort. Plus, they're much simpler than some of the more complicated functions you'll encounter later on. So, let's get this linear function figured out!

Calculating the Slope (m)

Alright, so how do we calculate the slope, m? The slope is the change in f(x) divided by the change in x. Think of it as "rise over run." We have two points: (1, 1) and (2, 5). So, the change in f(x) is 5 - 1 = 4, and the change in x is 2 - 1 = 1. Therefore, the slope m is 4 / 1 = 4.

The formula for the slope given two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1). In our case, (x1, y1) = (1, 1) and (x2, y2) = (2, 5). Plugging these values into the formula, we get m = (5 - 1) / (2 - 1) = 4 / 1 = 4. So, we've nailed down the slope! The line goes up 4 units for every 1 unit we move to the right.

Now that we know the slope, our function looks like this: f(x) = 4x + b. We're halfway there! We just need to find the y-intercept, b.

Understanding the slope is crucial because it tells us the rate at which the function is changing. In this case, for every increase of 1 in x, the value of the function increases by 4. This constant rate of change is a defining characteristic of linear functions. A steeper slope indicates a faster rate of change, while a gentler slope indicates a slower rate of change. Keep this concept in mind as you work with linear functions, and you'll be well on your way to mastering them!

Finding the Y-Intercept (b)

Okay, now for the y-intercept, b. We know that f(x) = 4x + b, and we also know that f(1) = 1. We can use this information to solve for b. Just plug in x = 1 and f(x) = 1 into the equation:

1 = 4(1) + b 1 = 4 + b

Now, subtract 4 from both sides: b = 1 - 4 = -3

So, the y-intercept b is -3. This means the line crosses the y-axis at the point (0, -3).

Alternatively, we could have used the other point, f(2) = 5. Plugging in x = 2 and f(x) = 5 into the equation f(x) = 4x + b, we get:

5 = 4(2) + b 5 = 8 + b

Subtracting 8 from both sides: b = 5 - 8 = -3.

We get the same answer for b using either point, which is a good sign! It confirms that our slope is correct. The y-intercept, b, is the value of the function when x is zero. It's an important point because it tells us where the line starts on the y-axis. Knowing the y-intercept and the slope gives us a complete picture of the linear function.

The Final Formula

Alright, we've got everything we need! We found that the slope m = 4 and the y-intercept b = -3. So, the formula for the linear function is:

f(x) = 4x - 3

That's it! We did it! This is the equation that describes the line that passes through the points (1, 1) and (2, 5).

To check our work, let's plug in x = 1 and x = 2 into our formula:

For x = 1: f(1) = 4(1) - 3 = 4 - 3 = 1. This matches the given information. For x = 2: f(2) = 4(2) - 3 = 8 - 3 = 5. This also matches the given information.

Since our formula works for both given points, we can be confident that we've found the correct linear function.

Conclusion

So, to recap, we started with two points on a line, used them to calculate the slope, then used the slope and one of the points to find the y-intercept. Finally, we combined the slope and y-intercept to write the equation of the linear function. Finding the formula of a linear function is like solving a puzzle, using the clues provided to reveal the hidden equation. It's a fundamental skill in algebra that opens the door to understanding more complex mathematical concepts.

Linear functions are essential tools in mathematics and various real-world applications. Understanding how to determine their formulas from given data points is a valuable skill. By following the steps outlined above, you can confidently tackle similar problems and gain a deeper appreciation for the power and versatility of linear functions. Great job, everyone! Keep practicing, and you'll become a linear function pro in no time!