Hey guys! Today, we're diving into the world of functions, specifically focusing on evaluating a quadratic function. We'll be working with the function g(x) = x² + 4x, and our mission is to find the values of this function at three different points: g(5), g(-5), and g(0). Think of it like this: we're plugging in these x-values into the function and seeing what y-values pop out! It's a fundamental concept in algebra, and mastering it opens the door to understanding more complex mathematical ideas. So, let's get started and break down each evaluation step-by-step.
a. Evaluating g(5)
Okay, let's tackle the first part: evaluating g(5). This means we're going to substitute x with 5 in our function, g(x) = x² + 4x. So, wherever we see an 'x', we're replacing it with a '5'. This gives us g(5) = (5)² + 4(5). Remember the order of operations (PEMDAS/BODMAS)? We need to handle the exponent first. 5 squared (5²) is 5 multiplied by itself, which equals 25. Now our expression looks like this: g(5) = 25 + 4(5). Next up is the multiplication: 4 multiplied by 5 is 20. So, now we have g(5) = 25 + 20. Finally, we add 25 and 20 together, and we get 45. Therefore, g(5) = 45. It's like we've found a point on the graph of the function – the point (5, 45). Understanding how to do this is crucial for all sorts of math problems, from graphing functions to solving equations. We're essentially finding the y-value that corresponds to the x-value of 5 on the graph of this function. This process of substitution and simplification is a cornerstone of function evaluation, and it’s something you'll use again and again in your math journey. The key is to be meticulous with your calculations and to follow the order of operations. Don't rush through the steps, and double-check your work to make sure you haven't made any simple arithmetic errors. With a little practice, you'll become a pro at evaluating functions in no time!
b. Evaluating g(-5)
Now, let's move on to evaluating g(-5). This time, we're substituting 'x' with -5 in our function, g(x) = x² + 4x. So, we get g(-5) = (-5)² + 4(-5). This is where things can get a little tricky with negative numbers, so we need to be extra careful. First, let's deal with the exponent: (-5)² means -5 multiplied by -5. A negative number multiplied by a negative number results in a positive number, so (-5)² equals 25. Our expression now looks like this: g(-5) = 25 + 4(-5). Next, we handle the multiplication: 4 multiplied by -5 is -20. Remember, a positive number multiplied by a negative number gives us a negative result. So, we have g(-5) = 25 + (-20). Adding a negative number is the same as subtracting, so this becomes g(-5) = 25 - 20. Finally, subtracting 20 from 25 gives us 5. Therefore, g(-5) = 5. It's important to pay close attention to the signs when dealing with negative numbers. One small mistake can throw off the entire calculation. Think of this as finding another point on the graph of our function, this time the point (-5, 5). This highlights an interesting aspect of quadratic functions – they can often have different y-values for positive and negative x-values of the same magnitude. This is due to the squared term, which always results in a positive value, regardless of the sign of the input. Practice makes perfect when it comes to working with negative numbers, so don't be afraid to tackle more problems like this. The more you practice, the more comfortable you'll become with the rules and the less likely you are to make mistakes.
c. Evaluating g(0)
Finally, let's evaluate g(0). This is often the easiest one! We're substituting 'x' with 0 in our function, g(x) = x² + 4x. So, we have g(0) = (0)² + 4(0). Zero squared (0²) is simply 0 multiplied by itself, which is 0. And 4 multiplied by 0 is also 0. So, our expression becomes g(0) = 0 + 0. Adding 0 and 0 gives us 0. Therefore, g(0) = 0. This tells us that the graph of the function passes through the origin (0, 0). Evaluating a function at x = 0 often gives us the y-intercept of the graph, which is the point where the graph crosses the y-axis. In this case, the y-intercept is 0. When you see g(0), it's a great opportunity to simplify the calculation quickly, as any term multiplied by 0 will become 0. This can save you a lot of time and effort. It's a fundamental property of zero that makes it so important in mathematics. Recognizing these simple cases can make evaluating functions much easier. Remember, understanding the behavior of a function at specific points, like x = 0, can give you valuable insights into the overall nature of the function and its graph. This is a key concept in understanding functions and their applications.
In conclusion, we've successfully evaluated the function g(x) = x² + 4x at x = 5, x = -5, and x = 0. We found that g(5) = 45, g(-5) = 5, and g(0) = 0. By carefully substituting the values and following the order of operations, we were able to determine the corresponding y-values for each given x-value. This process of function evaluation is a fundamental skill in algebra and is essential for understanding and working with functions in various mathematical contexts. Keep practicing, and you'll become a function evaluation master!