Geometric Sequence & Function Graphing: Math Problems Solved

by SLV Team 61 views

Hey guys! Today, we're diving into some cool math problems involving geometric sequences and function graphing. We'll break down each step, making sure everything's crystal clear. Let's get started!

1. Unlocking the Geometric Sequence: Finding the nth Term and the Sixth Term

So, our first task is to tackle a geometric sequence. We've got the sequence 1024, 512, 256, 128, and our mission is twofold: first, we need to figure out the explicit equation for the nth term, and second, we need to pinpoint what the sixth term in this sequence is. No sweat, right? Let's break it down step by step.

Understanding Geometric Sequences

First off, let's chat about what a geometric sequence actually is. In simple terms, it's a sequence where each term is found by multiplying the previous term by a constant number. This constant multiplier is what we call the common ratio. Identifying this common ratio is the key to unlocking the secrets of the sequence.

In our case, we can find the common ratio by dividing any term by its preceding term. For instance, 512 divided by 1024 gives us 1/2, or 0.5. Similarly, 256 divided by 512 also gives us 1/2. So, our common ratio, which we often denote as 'r', is 1/2. Now we're cooking with gas!

Crafting the Explicit Equation

Now that we know the common ratio, we can build the explicit equation. The general form for the nth term (often written as an) of a geometric sequence looks like this:

an = a1 * r^(n-1)

Where:

  • an is the nth term we want to find.
  • a1 is the first term of the sequence.
  • r is the common ratio.
  • n is the term number we're interested in.

In our sequence, a1 (the first term) is 1024, and r (the common ratio) is 1/2. Plugging these values into our general formula, we get:

an = 1024 * (1/2)^(n-1)

Boom! There's our explicit equation. This equation is like a magic formula that allows us to calculate any term in the sequence just by plugging in the term number 'n'. Pretty neat, huh?

Hunting Down the Sixth Term

Now, let's use our shiny new equation to find the sixth term. This means we need to find a6. To do this, we simply substitute n with 6 in our equation:

a6 = 1024 * (1/2)^(6-1) a6 = 1024 * (1/2)^5 a6 = 1024 * (1/32) a6 = 32

So, the sixth term of the sequence is 32. We've conquered the geometric sequence! We found the explicit equation and pinpointed the sixth term. Give yourselves a pat on the back, guys!

2. Graphing the Function y = (1/2)x: Unveiling the Y-Intercept, Domain, and Range

Alright, let's switch gears and dive into the world of function graphing. Our mission now is to graph the function y = (1/2)x, and while we're at it, we'll identify the y-intercept and define the domain and range. This might sound like a lot, but trust me, we'll break it down into manageable pieces.

Plotting the Points: The Key to Graphing

To graph this function, the most straightforward approach is to plot some points. We can do this by choosing a few x-values, plugging them into the equation, and calculating the corresponding y-values. These x and y pairs will give us coordinates that we can plot on a graph.

Let's pick a few values for x, say -2, -1, 0, 1, and 2. Now we'll plug each of these into our equation y = (1/2)x and see what we get:

  • If x = -2, then y = (1/2)^(-2) = 4
  • If x = -1, then y = (1/2)^(-1) = 2
  • If x = 0, then y = (1/2)^(0) = 1
  • If x = 1, then y = (1/2)^(1) = 1/2
  • If x = 2, then y = (1/2)^(2) = 1/4

So, we have the following points to plot: (-2, 4), (-1, 2), (0, 1), (1, 1/2), and (2, 1/4). If you plot these points on a graph, you'll see they form a curve. This is characteristic of an exponential function.

Spotting the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. In simpler terms, it's the y-value when x is equal to 0. We actually already calculated this! When x = 0, y = 1. So, our y-intercept is 1. This means the graph crosses the y-axis at the point (0, 1). Identifying the y-intercept gives us a crucial anchor point for our graph.

Deciphering the Domain and Range

Now, let's talk about the domain and range. These terms describe the set of all possible x-values (domain) and y-values (range) that the function can take.

  • Domain: For the function y = (1/2)x, we can plug in any real number for x. There are no restrictions. So, the domain is all real numbers. We can write this as (-∞, ∞) in interval notation.
  • Range: The range is a little different. Exponential functions of this form will always produce positive y-values. The function will get closer and closer to 0 as x gets larger, but it will never actually reach 0. So, the range is all positive real numbers. In interval notation, we write this as (0, ∞).

So, to recap, we've graphed the function y = (1/2)x, identified the y-intercept as 1, and determined the domain to be all real numbers and the range to be all positive real numbers. Awesome work!

Wrapping It Up: Math Problem Solved!

And there you have it, guys! We've successfully navigated through both a geometric sequence problem and a function graphing challenge. We figured out the explicit equation for the nth term of the sequence, found the sixth term, graphed the function, identified the y-intercept, and nailed down the domain and range. You're all math superstars!

Remember, the key to tackling math problems is to break them down into smaller, more manageable steps. Understand the core concepts, apply the right formulas, and don't be afraid to ask questions. Keep practicing, and you'll become math whizzes in no time! Keep your curiosity burning, and stay tuned for more math adventures!