Finding The Equation Of A Linear Function F(3)=-13 And F(10)=-48

Hey guys! Let's dive into how to find the equation of a linear function when given two points. This is a common problem in algebra, and mastering it will definitely boost your math skills. In this article, we'll break down the steps using a real example. So, let's get started!

Understanding Linear Functions

Before we jump into the problem, let’s quickly recap what a linear function is. A linear function is a function that forms a straight line when graphed. The general form of a linear function is f(x) = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope m tells us how steep the line is, and the y-intercept b is the point where the line crosses the y-axis. Understanding this basic form is crucial for tackling any problem involving linear functions.

When we're given two points, our main goal is to find the values of m and b. Once we have these, we can write the equation of the line. So, how do we find these values? That's what we'll explore in the next section. Remember, linear functions are all about straight lines, so think visually! Each point we're given helps us draw that line and figure out its unique equation. We can calculate the slope using the two points, which gives us a sense of the line's direction and steepness. The y-intercept then anchors the line on the graph. By finding these two key values, we completely define the linear function. So, keep the form f(x) = mx + b in mind as we move forward; it's the key to unlocking these problems. And don't worry, we'll break down each step so it feels super clear and manageable.

Problem Setup

Now, let's look at the specific problem we're trying to solve. We're given two points defined by the function f(x). These points are f(3) = -13 and f(10) = -48. What this means is that when x is 3, the value of the function (or y) is -13. Similarly, when x is 10, the value of the function is -48. We can think of these as two coordinate pairs: (3, -13) and (10, -48). These are the two points our line passes through, and they're our starting blocks for finding the equation. To make things even clearer, let's visualize these points on a graph. Imagine a coordinate plane, and plot these two points. You'll see that they form a line when connected. Our task is to find the equation that describes this line perfectly. So, we need to find that slope and y-intercept we talked about earlier. With these two points in hand, we can start calculating. The next step involves using the slope formula, which is our main tool for finding the steepness of the line. So, hold onto these coordinate pairs; we're about to put them to work!

The goal is to find the equation in the form f(x) = mx + b, where we need to determine the values of m (the slope) and b (the y-intercept). This is a classic setup for a linear function problem. We have two points, and from these two points, we can extract all the information we need to define the line. It's like having two pieces of a puzzle; we just need to fit them together to see the whole picture. So, keep in mind that our ultimate goal is to plug in values for m and b into that equation f(x) = mx + b. This is the template we're aiming to fill. Each step we take, from calculating the slope to finding the y-intercept, is a step towards completing this equation. So, let's move on to the next stage, where we'll start crunching some numbers!

Step 1: Calculate the Slope (m)

The slope (m) of a line is a measure of its steepness and direction. It tells us how much the y-value changes for every unit change in the x-value. To calculate the slope, we use the formula: m = (y2 - y1) / (x2 - x1). Here, (x1, y1) and (x2, y2) are the coordinates of the two points we have. In our case, we have the points (3, -13) and (10, -48). Let’s assign these values: x1 = 3, y1 = -13, x2 = 10, and y2 = -48. Now, we can plug these values into the slope formula:

m = (-48 - (-13)) / (10 - 3)

Simplifying the numerator, we get:

m = (-48 + 13) / (10 - 3)

m = -35 / 7

Finally, dividing -35 by 7, we find the slope:

m = -5

So, the slope of our line is -5. This means that for every 1 unit increase in x, the y-value decreases by 5 units. A negative slope indicates that the line slopes downward from left to right. Now that we have the slope, we're halfway to finding the equation of the line. The slope is a crucial piece of information; it tells us the rate of change of our function. It's like the engine that drives the line's direction. With the slope in hand, we can move on to the next step, which is finding the y-intercept.

Step 2: Find the Y-intercept (b)

The y-intercept (b) is the point where the line crosses the y-axis. It's the value of y when x is 0. To find the y-intercept, we can use the slope-intercept form of the equation (f(x) = mx + b) and plug in the slope we just calculated and one of the given points. We have the slope m = -5, and let’s use the point (3, -13). Plugging these values into the equation, we get:

-13 = (-5)(3) + b

Now, we solve for b:

-13 = -15 + b

To isolate b, we add 15 to both sides of the equation:

-13 + 15 = b

2 = b

So, the y-intercept b is 2. This means the line crosses the y-axis at the point (0, 2). The y-intercept is like the anchor point of the line; it fixes the line's vertical position. Now that we have both the slope and the y-intercept, we have all the pieces we need to write the equation of the line. The next step is simply putting it all together. We could have used the other point (10, -48) as well, and we would have gotten the same result for b. It's a good idea to try it out to confirm your understanding! So, let's move on to the final step and write out the equation.

Step 3: Write the Equation

Now that we have the slope m = -5 and the y-intercept b = 2, we can write the equation of the linear function in the form f(x) = mx + b. We simply plug in the values we found for m and b:

f(x) = -5x + 2

This is the equation of the linear function that passes through the points (3, -13) and (10, -48). It tells us everything we need to know about this line. For any value of x, we can plug it into this equation and find the corresponding value of f(x). This equation is like the DNA of the line; it uniquely identifies it. To check our answer, we can plug in the x-values from our original points and see if we get the correct y-values. For example, let's check with x = 3:

f(3) = -5(3) + 2

f(3) = -15 + 2

f(3) = -13

This matches the given value, so our equation is correct! We can do the same check with x = 10 to be even more confident. So, there you have it! We've successfully found the equation of the linear function. We started with two points, calculated the slope, found the y-intercept, and then put it all together into the equation. It's a step-by-step process that you can use for any linear function problem.

Final Answer

So, the equation of the linear function is:

f(x) = -5x + 2

We found that the missing values are -5 for the slope and 2 for the y-intercept. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. First, find the slope using the slope formula. Second, use the slope and one of the points to find the y-intercept. Finally, plug the slope and y-intercept into the slope-intercept form of the equation. And always remember to double-check your work by plugging in your original points! Linear functions are a fundamental concept in algebra, and mastering them will open the door to more advanced topics. Keep practicing, and you'll become a pro in no time!

Conclusion

Finding the equation of a linear function given two points might seem tricky at first, but by following these steps, you can solve any similar problem. We started by understanding the basics of linear functions, then set up the problem, calculated the slope, found the y-intercept, and finally wrote the equation. This methodical approach is key to success in math. Remember, practice makes perfect, so keep working on these types of problems, and you'll become more confident. Linear functions are a building block for many other mathematical concepts, so the time you invest in understanding them now will pay off in the long run. Keep up the great work, and happy problem-solving!