Ordered Pair On Graph F(x) = X + 4 Where X = 3p

by Sebastian Müller 48 views

Hey math enthusiasts! Let's dive into a fun problem that involves understanding functions and ordered pairs. We're given the function f(x) = x + 4 and asked to find the ordered pair representing a specific point on the graph where x = 3p. Don't worry, it's not as complicated as it sounds! We'll break it down step by step, making sure everyone gets a clear picture of what's going on.

Understanding the Basics: Functions and Ordered Pairs

Before we jump into the solution, let's quickly refresh our understanding of functions and ordered pairs. Think of a function like a machine: you input something (x in this case), and the machine processes it according to a specific rule (f(x) = x + 4), and then it outputs something else. In our function, the rule is simple: take the input x and add 4 to it. The result of this operation is the output, which we often represent as y or f(x).

An ordered pair is simply a way to represent a point on a graph. It consists of two numbers, the x-coordinate and the y-coordinate, written in parentheses like this: (x, y). The x-coordinate tells us how far to move horizontally from the origin (the point where the axes cross), and the y-coordinate tells us how far to move vertically. When we graph a function, we're essentially plotting a series of ordered pairs that satisfy the function's rule. Each point (x, f(x)) on the graph represents a specific input x and its corresponding output f(x).

In this problem, we're given a specific input, x = 3p, and we need to find the corresponding output f(3p). This will give us the y-coordinate of our ordered pair. Once we have both the x-coordinate and the y-coordinate, we can write the ordered pair in the form (x, y) or (3p, f(3p)). Let's move on to finding that y-coordinate!

Solving for the Ordered Pair: When x = 3p

Okay, now for the fun part: plugging in x = 3p into our function f(x) = x + 4. This is where the magic happens! Remember, whatever we see an x in the function, we're going to replace it with 3p. So, here we go:

f(3p) = (3p) + 4

That's it! We've successfully substituted 3p for x. Now, let's think about what this means. The expression 3p + 4 represents the output of our function when the input is 3p. It's the y-coordinate of the point on the graph where x = 3p. We can't simplify this expression any further without knowing the value of p, so we'll leave it as 3p + 4.

So, what's our ordered pair? We know the x-coordinate is 3p, and we just found that the y-coordinate is 3p + 4. Therefore, the ordered pair is (3p, 3p + 4). This ordered pair represents a specific point on the graph of the function f(x) = x + 4. The location of this point will depend on the value of p. For example, if p = 0, the point would be (0, 4). If p = 1, the point would be (3, 7). And so on.

See? It wasn't so bad after all! By understanding the basics of functions and ordered pairs, and by carefully substituting x = 3p into the function, we were able to find the ordered pair representing the point on the graph. Let's celebrate that!

Evaluating the Answer Choices

Now that we've solved the problem, let's take a look at the answer choices provided and see which one matches our solution. We found that the ordered pair is (3p, 3p + 4). Let's examine the options:

A. (x, x + 4): This option represents the general form of an ordered pair for the function f(x) = x + 4, but it doesn't specifically address the case where x = 3p. It's a correct representation of the function's points in general, but not the specific solution we're looking for.

B. (x, 3p + 4): This option has the correct y-coordinate (3p + 4), but it uses x as the x-coordinate. We know that the x-coordinate should be 3p, so this option is incorrect.

C. (3p, x + 4): This option has the correct x-coordinate (3p), but the y-coordinate (x + 4) is not in terms of p. We need the y-coordinate to be the result of plugging 3p into the function, which we found to be 3p + 4. So, this option is also incorrect.

D. (3p, 3p + 4): This option perfectly matches our solution! It has the correct x-coordinate (3p) and the correct y-coordinate (3p + 4). This is the ordered pair that represents the point on the graph of f(x) = x + 4 where x = 3p.

Therefore, the correct answer is D. We've not only found the solution but also carefully evaluated each answer choice to confirm our result. That's how you ace these problems, guys! Keep up the awesome work!

Key Takeaways and Tips

Before we wrap up, let's highlight some key takeaways and tips that you can apply to similar problems in the future:

  • Understand the Basics: Make sure you have a solid grasp of the definitions of functions, ordered pairs, and how they relate to graphing. This will form the foundation for solving more complex problems.
  • Substitution is Key: When you're given a specific value for x (or any variable), the key is to substitute that value into the function's equation. This will give you the corresponding y-coordinate.
  • Pay Attention to Notation: Ordered pairs are always written in the form (x, y). Make sure you keep the x-coordinate and y-coordinate in the correct order.
  • Evaluate Answer Choices Carefully: Don't just pick the first answer that looks right. Take the time to evaluate each option and make sure it truly matches your solution.
  • Practice Makes Perfect: The more you practice solving these types of problems, the more comfortable and confident you'll become. Seek out similar examples and work through them step by step.

Remember, math isn't about memorizing formulas; it's about understanding concepts and applying them in different situations. By breaking down problems into smaller steps, and by focusing on the underlying principles, you can conquer any math challenge that comes your way. You've got this!

So, the next time you encounter a problem involving functions and ordered pairs, remember the steps we've discussed here. Understand the basics, substitute carefully, pay attention to notation, evaluate answer choices, and most importantly, practice! With these tools in your arsenal, you'll be well-equipped to tackle any math problem with confidence.

Keep exploring, keep learning, and keep having fun with math! And remember, there's always something new and exciting to discover in the world of mathematics. Happy problem-solving, everyone!