Hey guys! Let's break down this math problem together. We're going to simplify the function h(x) = (8^x * 8^7) / 8^10 and then figure out how its graph relates to the graph of the simpler function p(x) = 8^x. Think of it as a mathematical makeover – we're taking a slightly complex function and turning it into something sleek and easy to understand. Plus, we'll see how this transformation looks graphically. Ready to dive in?
Simplifying the Function h(x) = (8^x * 8^7) / 8^10
Okay, our first step is to simplify the function h(x) = (8^x * 8^7) / 8^10. This might look a little intimidating at first, but don't worry, we've got this! The key here is to remember the rules of exponents. These rules are like the secret sauce that makes simplifying these kinds of expressions a breeze.
Specifically, we're going to use two main exponent rules:
- Product of Powers: When you're multiplying exponential expressions with the same base, you add the exponents. Mathematically, this looks like a^m * a^n = a^(m+n). This means that if we have something like 8^x * 8^7, we can combine those into a single term with the base 8 and a new exponent that's the sum of x and 7.
- Quotient of Powers: When you're dividing exponential expressions with the same base, you subtract the exponents. This rule is expressed as a^m / a^n = a^(m-n). So, if we have 8^(some exponent) / 8^10, we can simplify this by subtracting 10 from the exponent in the numerator.
Let's apply these rules step-by-step. First, we'll focus on the numerator of our function, which is 8^x * 8^7. Using the Product of Powers rule, we can combine these terms:
8^x * 8^7 = 8^(x + 7)
Great! Now our function looks like this:
h(x) = 8^(x + 7) / 8^10
Next up, we need to deal with the division. This is where the Quotient of Powers rule comes in handy. We're dividing 8^(x + 7) by 8^10, so we subtract the exponents:
h(x) = 8^((x + 7) - 10)
Now, let's simplify the exponent by combining the constants:
h(x) = 8^(x + 7 - 10) h(x) = 8^(x - 3)
And there you have it! We've successfully simplified the function. The simplified form of h(x) is 8^(x - 3). See? It wasn't so scary after all. By applying the rules of exponents, we transformed a seemingly complex expression into a much more manageable one. This simplified form will make it much easier to analyze the function's behavior and understand its graph, which is exactly what we'll do in the next section. Understanding these exponent rules is crucial for simplifying exponential functions, and with practice, you'll be able to apply them quickly and confidently. So, keep practicing, and you'll become an exponent pro in no time!
Describing the Transformation from p(x) = 8^x to h(x) = 8^(x-3)
Now that we've simplified h(x) to 8^(x - 3), the next step is to figure out how its graph relates to the graph of p(x) = 8^x. This is where the concept of graph transformations comes into play. Graph transformations are like visual modifications – they shift, stretch, compress, or reflect a graph, changing its position and shape in the coordinate plane. Understanding these transformations allows us to quickly sketch and analyze graphs without having to plot a ton of points.
In our case, we're specifically dealing with a horizontal translation. A horizontal translation shifts the graph left or right along the x-axis. The general form for a horizontal translation is f(x - c), where c is the amount of the shift. If c is positive, the graph shifts to the right, and if c is negative, the graph shifts to the left. Think of it as the x value being "delayed" or "advanced."
Looking at our functions, we have p(x) = 8^x and h(x) = 8^(x - 3). Notice that h(x) has the form f(x - c), where f(x) = 8^x and c = 3. Since c is positive, this tells us that the graph of h(x) is a horizontal translation of the graph of p(x) to the right. But how much to the right? Well, the value of c tells us the amount of the shift. In this case, c = 3, so the graph of h(x) is shifted 3 units to the right compared to the graph of p(x).
To visualize this, imagine taking the graph of p(x) = 8^x and simply sliding it 3 units to the right along the x-axis. That's exactly what the transformation does! Every point on the graph of p(x) gets moved 3 units to the right to create the graph of h(x). This horizontal shift doesn't change the basic shape of the exponential curve; it just repositions it on the coordinate plane.
For example, the point (0, 1) on the graph of p(x) = 8^x will be shifted to the point (3, 1) on the graph of h(x) = 8^(x - 3). Similarly, the point (1, 8) on p(x) will be shifted to (4, 8) on h(x), and so on. The y-values remain the same, while the x-values are increased by 3.
In summary, the function h(x) = 8^(x - 3) represents a horizontal translation of the function p(x) = 8^x by 3 units to the right. Understanding this transformation allows us to quickly sketch the graph of h(x) by simply shifting the graph of p(x). This is a powerful technique for analyzing and understanding functions, and it's a key concept in algebra and calculus. So, the next time you see a function in the form f(x - c), remember that it represents a horizontal translation, and you'll be able to visualize its graph with ease! Mastering these graph transformations will significantly enhance your ability to understand and work with functions.
Conclusion: Putting It All Together
Alright, guys, we've tackled quite a bit in this mathematical journey! We started with a slightly complex exponential function, h(x) = (8^x * 8^7) / 8^10, and simplified it using the power of exponent rules. We then connected this simplified form to a more basic function, p(x) = 8^x, by exploring the concept of graph transformations, specifically horizontal translations.
First, let's recap the simplification process. By applying the Product of Powers rule, we combined the terms in the numerator: 8^x * 8^7 = 8^(x + 7). Then, using the Quotient of Powers rule, we simplified the entire function: 8^(x + 7) / 8^10 = 8^((x + 7) - 10) = 8^(x - 3). So, we successfully simplified h(x) to 8^(x - 3). This simplification is not just about making the function look cleaner; it makes it much easier to analyze and understand its behavior.
Next, we delved into the world of graph transformations. We discovered that the function h(x) = 8^(x - 3) is a horizontal translation of the function p(x) = 8^x. Specifically, it's a translation 3 units to the right. This means that the graph of h(x) is simply the graph of p(x) shifted 3 units along the positive x-axis. Every point on the original graph moves 3 units to the right, maintaining the same vertical position. This understanding allows us to visualize the graph of h(x) without having to plot individual points, which is a significant time-saver and a powerful tool for understanding functions. The ability to recognize and apply graph transformations is a crucial skill in mathematics, particularly in algebra and calculus.
So, what are the key takeaways from this exploration? We've learned that simplifying expressions using exponent rules can make them much easier to work with. We've also seen how graph transformations can provide valuable insights into the relationship between functions and their graphs. By combining these techniques, we can gain a deeper understanding of mathematical concepts and solve problems more efficiently. This combination of algebraic manipulation and graphical analysis is a powerful approach to problem-solving in mathematics.
Remember, math isn't just about memorizing formulas and rules; it's about understanding the underlying concepts and how they connect. By breaking down problems into smaller steps, applying the right tools, and thinking critically, you can conquer even the most challenging mathematical tasks. So, keep practicing, keep exploring, and keep asking questions! You've got this!
