Ordering Functions By Minimum Value: A Math Guide

Hey math enthusiasts! Today, we're diving into a fun little puzzle: ordering functions based on their minimum values. We'll be working with a quadratic function and a function presented as a table of values. Sounds like a blast, right? Let's break it down step by step so you can ace this type of problem.

Understanding Minimum Values

Before we jump into the functions themselves, let's make sure we're all on the same page about what a minimum value actually is. In simple terms, the minimum value of a function is the lowest point the function reaches. Think of it like the bottom of a valley on a graph. For a quadratic function (like our f(x)), this minimum value occurs at the vertex of the parabola. For a function represented by a table, we need to carefully look through the output values (the g(x) values in our case) to find the smallest one.

Why is understanding minimum values important? Well, in many real-world applications, finding the minimum (or maximum) value of a function can be super useful. Imagine you're trying to minimize costs in a business, or maximize the height of a projectile. Knowing how to find these extreme values is a key skill in mathematics and beyond. So, let's get to it!

Function f(x) = 2x² - 8x + 1: Finding the Minimum Value

Okay, let's tackle our first function: f(x) = 2x² - 8x + 1. This is a quadratic function, which means its graph is a parabola. Since the coefficient of the x² term (the '2' in front) is positive, the parabola opens upwards, meaning it has a minimum point. To find this minimum, we need to find the vertex of the parabola. There are a couple of ways we can do this. One popular method is to complete the square. This involves rewriting the quadratic expression in vertex form, which directly reveals the vertex coordinates. Another method, and the one we'll use here, is to use the formula for the x-coordinate of the vertex: x = -b / 2a. Remember those coefficients from the quadratic formula? They're back! In our function, a = 2 and b = -8. Plugging these values into the formula, we get:

x = -(-8) / (2 * 2) = 8 / 4 = 2

So, the x-coordinate of the vertex is 2. To find the y-coordinate (which is our minimum value), we plug this x-value back into the original function:

f(2) = 2(2)² - 8(2) + 1 = 2(4) - 16 + 1 = 8 - 16 + 1 = -7

Ta-da! The minimum value of f(x) is -7. That wasn't so bad, was it? We've successfully navigated the quadratic terrain and pinpointed the lowest point of our first function. Now, let's move on to the next challenge: deciphering the minimum value from a table of values.

Function g(x): Minimum Value from a Table

Now, let's shift our focus to the function g(x), which is presented to us as a table of values. This is a different kind of beast compared to our quadratic friend f(x). Instead of a neat equation, we have a collection of x values and their corresponding g(x) values. Finding the minimum value here is a bit like a scavenger hunt – we need to carefully scan the g(x) values and find the smallest one. Let's take a look at the table again:

Alright, our mission is clear: find the smallest g(x) value in this table. As we scan through the g(x) column, we see 3, -1, -3, -3, -1, and 3. Which one is the smallest? It's -3! Notice that -3 appears twice in the table, at x = -2 and x = -1. This means that the function g(x) reaches its minimum value of -3 at two different points. Pretty interesting, huh? So, we've successfully identified the minimum value of g(x) from the table. Now, the final showdown awaits: comparing the minimum values of f(x) and g(x).

Comparing the Minimum Values

We've reached the final stage of our mathematical journey! We've found the minimum value of f(x) to be -7, and the minimum value of g(x) to be -3. Now, the task is simple: which one is smaller? Think of a number line. Numbers further to the left are smaller. So, is -7 to the left or right of -3? It's to the left! This means that -7 is smaller than -3.

Therefore, the minimum value of f(x) is smaller than the minimum value of g(x). We can confidently say that f(x) has a smaller minimum value. Now, let's put it all together and give our final answer in a clear and concise way.

Conclusion

Alright, guys, we've done it! We've successfully navigated the world of functions and their minimum values. We started by understanding what minimum values are, then we found the minimum value of the quadratic function f(x) using the vertex formula, and we hunted down the minimum value of g(x) from a table of values. Finally, we compared these minimum values and concluded that the minimum value of f(x) is smaller than the minimum value of g(x). In summary, ordering the functions from smallest to largest minimum value, we have:

1. f(x) (minimum value: -7)
2. g(x) (minimum value: -3)

So there you have it! You've not only solved the problem but also gained a deeper understanding of minimum values and how to find them. Keep practicing, and you'll be a master of functions in no time!