Evaluating (1/3)h(-8) For H(x) = X^2 - 5x + 7

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate the expression (1/3) * h(-8), given that h(x) is a quadratic function defined as h(x) = x^2 - 5x + 7. This might sound a bit intimidating at first, but trust me, it's super manageable once we break it down step by step. We'll go through the process together, making sure each step is crystal clear. So, grab your calculators (or your brainpower!) and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we fully understand what the problem is asking. We're given a function, h(x) = x^2 - 5x + 7, which is a quadratic function. A quadratic function, as you might remember, is a polynomial function of degree two. This means the highest power of x in the function is 2. The graph of a quadratic function is a parabola, which is a U-shaped curve. But for this problem, we don't need to worry about the graph; we're just focusing on evaluating the function for a specific input.

The expression we need to evaluate is (1/3) * h(-8). This means we first need to find the value of the function h when x is -8. In other words, we need to substitute -8 for x in the expression for h(x). Once we have that value, we'll multiply it by 1/3 to get our final answer. Think of it like this: we're taking one-third of the value of the function h at the point x = -8.

Breaking down the problem like this makes it much less daunting. We have a clear plan: first, find h(-8), and then multiply the result by 1/3. It’s all about tackling it one step at a time. So, let's move on to the first step: finding h(-8).

Step 1: Evaluating h(-8)

This is the heart of the problem, guys! To evaluate h(-8), we need to substitute -8 for every instance of x in the function h(x) = x^2 - 5x + 7. It’s crucial to be careful with the signs and the order of operations (PEMDAS/BODMAS) to avoid any mistakes. Trust me, even a small sign error can throw off the entire calculation, and we want to get this right.

So, let's do it together. We start with the function:

h(x) = x^2 - 5x + 7

Now, replace x with -8:

h(-8) = (-8)^2 - 5(-8) + 7

See what we did there? Every x has been carefully replaced with -8. Now, we need to simplify this expression following the order of operations. Remember, this means we handle exponents first, then multiplication and division (from left to right), and finally addition and subtraction (from left to right).

First, let's deal with the exponent: (-8)^2. This means -8 multiplied by itself:

(-8)^2 = (-8) * (-8) = 64

Remember that a negative number multiplied by a negative number gives a positive result. This is a common area for errors, so it’s always good to double-check. Next, we handle the multiplication: -5(-8). Again, we have a negative number multiplied by a negative number, so the result will be positive:

-5(-8) = 40

Now we can substitute these values back into our expression for h(-8):

h(-8) = 64 + 40 + 7

Finally, we add these numbers together:

h(-8) = 104 + 7 = 111

Awesome! We've found that h(-8) = 111. That's a significant step forward. We've successfully evaluated the function h at x = -8. Now, we're ready for the final step: multiplying this result by 1/3.

Step 2: Multiplying by 1/3

Okay, guys, we're in the home stretch now! We've already figured out that h(-8) = 111. The final step is to multiply this value by 1/3, as the original expression we need to evaluate is (1/3) * h(-8). This is a straightforward multiplication, but it’s still important to do it carefully to ensure we get the correct final answer.

So, we have:

(1/3) * h(-8) = (1/3) * 111

Multiplying a number by 1/3 is the same as dividing it by 3. So, we can rewrite this as:

(1/3) * 111 = 111 / 3

Now, we just need to perform the division. You can do this using long division, a calculator, or mental math if you're feeling confident. Let’s go through it step by step to make sure we’re clear on the process. How many times does 3 go into 11? It goes in 3 times (3 * 3 = 9), leaving a remainder of 2. We bring down the next digit, which is 1, making the number 21. How many times does 3 go into 21? It goes in exactly 7 times (3 * 7 = 21), with no remainder.

So, 111 divided by 3 is 37:

111 / 3 = 37

Therefore:

(1/3) * h(-8) = 37

And that's it! We've successfully evaluated the expression (1/3) * h(-8). The final answer is 37. We made it through all the steps, from understanding the problem to the final calculation, and we got the correct result. Give yourselves a pat on the back, guys – you earned it!

Final Answer

So, to recap, we were asked to evaluate the expression (1/3) * h(-8) given that h(x) = x^2 - 5x + 7. We broke the problem down into two main steps:

1. Evaluating h(-8): We substituted -8 for x in the function h(x) and carefully calculated the result, finding that h(-8) = 111.
2. Multiplying by 1/3: We then multiplied the result from step 1 by 1/3, which is the same as dividing by 3. This gave us the final answer of 37.

Therefore, the final answer is:

(1/3) * h(-8) = 37

We did it! I hope this step-by-step explanation has been helpful and has made the process clear and understandable. Remember, math problems often seem daunting at first, but by breaking them down into smaller, manageable steps, we can tackle even the most challenging problems. Keep practicing, keep asking questions, and most importantly, keep having fun with math!