Finding The 1st-Degree Function Passing Through (0,5) And (6,3)

by Scholario Team 64 views

Hey guys! Let's dive into a super interesting math problem today. We're going to explore a 1st-degree function – you know, those linear functions we all love – and figure out its secrets based on some points it passes through. Specifically, we're looking at a function that goes from the set of real numbers (IR) to the set of real numbers (IR), and its graph includes the points (0, 5) and (6, 3). Sounds like a fun challenge, right? So, grab your thinking caps, and let's get started!

Understanding 1st-Degree Functions

First off, let's make sure we're all on the same page about what a 1st-degree function actually is. These functions are the simplest kind of polynomial functions, and they're also called linear functions. Why linear? Because when you graph them, they form a straight line. Pretty neat, huh? The general form of a 1st-degree function is usually written as:

f(x) = ax + b

Where:

  • f(x) is the value of the function at x (often also written as y)
  • x is the input variable
  • a is the slope of the line (how steep it is)
  • b is the y-intercept (where the line crosses the y-axis)

So, in our case, we're trying to find the specific values of a and b that make our function's graph pass through the points (0, 5) and (6, 3). Think of it like this: we have two clues, and we need to use them to solve the mystery of our function!

The slope, represented by 'a' in our equation, is a crucial characteristic of a linear function. It tells us how much the function's value changes for every unit change in 'x'. A positive slope means the line goes upwards as you move from left to right, while a negative slope means it goes downwards. The steeper the line, the larger the absolute value of the slope. Understanding the slope is fundamental to grasping the behavior of linear functions.

The y-intercept, represented by 'b', is another key element. It's the point where the line intersects the y-axis, which occurs when x = 0. In other words, it's the value of the function when the input is zero. The y-intercept gives us a starting point for visualizing the line on a graph. Knowing the y-intercept and the slope allows us to accurately sketch the graph of any linear function.

Using the Given Points

Okay, now we get to the juicy part – using the points (0, 5) and (6, 3) to figure out our function. Remember, each point gives us an x and a y value that must satisfy our equation f(x) = ax + b. So, let's plug them in and see what happens.

Point (0, 5)

This point is super helpful because it directly gives us the y-intercept. When x = 0, f(x) = 5. If we plug these values into our equation, we get:

5 = a * 0 + b

This simplifies to:

5 = b

Ta-da! We've found b! The y-intercept of our line is 5. That was easier than we thought, right? This makes sense because the y-intercept is, by definition, the point where the line crosses the y-axis (where x = 0), and we were given that point directly.

Point (6, 3)

Now that we know b = 5, we can use the other point, (6, 3), to find a. Let's plug x = 6, f(x) = 3, and b = 5 into our equation:

3 = a * 6 + 5

Now we have a simple equation to solve for a. Let's subtract 5 from both sides:

3 - 5 = 6a

-2 = 6a

And finally, divide both sides by 6:

a = -2 / 6

a = -1 / 3

Awesome! We've found a! The slope of our line is -1/3. This means that for every 3 units we move to the right on the graph (increase in x), the line goes down 1 unit (decrease in y). The negative slope tells us the line is sloping downwards.

The Function Revealed

Okay, guys, we've done it! We've found both a and b, which means we can write out the equation for our 1st-degree function. Remember, the general form is:

f(x) = ax + b

We found that a = -1/3 and b = 5. So, our specific function is:

f(x) = (-1/3)x + 5

This is the equation of the line that passes through the points (0, 5) and (6, 3). We can even double-check our answer by plugging in the x-values of our points and seeing if we get the correct y-values. Let's try it!

Checking with (0, 5)

f(0) = (-1/3) * 0 + 5

f(0) = 0 + 5

f(0) = 5

Yay! It works! When x = 0, f(x) = 5, just like our point told us.

Checking with (6, 3)

f(6) = (-1/3) * 6 + 5

f(6) = -2 + 5

f(6) = 3

Double yay! It works again! When x = 6, f(x) = 3. This confirms that our equation is correct.

Graphing the Function

To really understand our function, it's helpful to visualize it. We know it's a straight line, and we know two points on that line: (0, 5) and (6, 3). We can plot these points on a graph and then draw a line through them. The line will continue infinitely in both directions, representing all the possible input and output values of our function.

The y-intercept, which is 5 in our case, gives us a clear starting point on the y-axis. Then, using the slope of -1/3, we can imagine moving 3 units to the right and 1 unit down to find another point on the line. Connecting these points (or any two points on the line) gives us a visual representation of the function's behavior. Graphing helps us see how the function changes as x changes, and it provides a valuable tool for understanding linear relationships.

Real-World Applications

Okay, so we've solved this math problem, which is great! But you might be wondering, "Why does this matter? Where would I ever use this in real life?" Well, 1st-degree functions are actually super useful for modeling tons of real-world situations. Think about anything that has a constant rate of change – that's where linear functions come in handy.

For example:

  • Distance and Time: If you're driving at a constant speed, the distance you've traveled is a linear function of time. The slope would be your speed, and the y-intercept could be your starting position.
  • Cost and Quantity: The total cost of something might be a linear function of the number of items you buy, especially if there's a fixed cost plus a cost per item. The slope would be the cost per item, and the y-intercept would be the fixed cost.
  • Temperature Conversion: Converting between Celsius and Fahrenheit is a linear function. Each degree Celsius increase corresponds to a specific increase in Fahrenheit.
  • Simple Interest: The amount of money you have in a savings account with simple interest grows linearly over time. The slope is the interest rate, and the y-intercept is your initial deposit.

So, understanding 1st-degree functions isn't just about solving math problems – it's about understanding how things change in the world around us! By identifying linear relationships, we can make predictions, analyze data, and solve practical problems.

Conclusion

So, there you have it! We've successfully found the 1st-degree function that passes through the points (0, 5) and (6, 3). We used our knowledge of linear functions, plugged in the given points, solved for the slope and y-intercept, and even checked our answer. We also explored how these functions can be used to model real-world situations. Not bad for a day's work, huh? Remember, guys, math isn't just about numbers and equations – it's about understanding patterns and relationships. And 1st-degree functions are a fantastic place to start!

I hope you enjoyed this little math adventure! Keep practicing, keep exploring, and keep those brain cells firing! You got this!