Range Of F(x) = -(x+3)^2 + 7? Solve It Now!

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Range of the Function f(x) = -(x+3)^2 + 7? Let's Solve It!

Hey guys! Today, we're diving into a super interesting math problem: finding the range of the function f(x) = -(x+3)^2 + 7. This might seem a bit intimidating at first, but trust me, we'll break it down step by step so it’s super easy to understand. We will explore the range of quadratic functions, particularly focusing on how the different components of the function affect its output. By the end of this guide, you'll not only know the answer but also understand why it's the answer. So, grab your thinking caps, and let's get started!

Understanding the Function

Okay, first things first, let's really understand what the function f(x) = -(x+3)^2 + 7 is telling us. This is a quadratic function, which basically means it's a polynomial function where the highest power of x is 2. Quadratic functions have a distinctive U-shape when graphed, which we call a parabola. Now, the key here is to notice a few things about our specific function:

  • The Negative Sign: See that negative sign in front of the (x+3)^2 term? That's a big clue! It tells us that the parabola opens downwards. Think of it like a frown instead of a smile. This means the function has a maximum value, a highest point, rather than a minimum.
  • The (x+3)^2 term: This part tells us about the horizontal shift of the parabola. The “+3” inside the parentheses actually shifts the parabola 3 units to the left along the x-axis. Remember, it's the opposite of what you might initially think!
  • The +7: This is the vertical shift. It moves the entire parabola 7 units up along the y-axis. This is super important because it directly affects the maximum value the function can reach.

So, to recap, we've got a downward-opening parabola that's been shifted 3 units left and 7 units up. Understanding these transformations is crucial for figuring out the range. We'll see how these shifts impact the maximum value of our function and, consequently, its range. These transformations are not just abstract concepts; they visually and mathematically shape the function's behavior, giving us key insights into its range. By understanding each component, we can piece together the overall picture of what the function does and where its output values lie.

Finding the Maximum Value

Now, let's zoom in on finding the maximum value of our function, which is super important for figuring out the range. Remember how we talked about the parabola opening downwards? Well, that means it has a highest point, called the vertex. The y-coordinate of this vertex is the maximum value of the function. For our function, f(x) = -(x+3)^2 + 7, we can actually find this vertex directly from the equation. The vertex form of a quadratic equation is f(x) = a(x-h)^2 + k, where (h, k) is the vertex.

Comparing this general form to our specific function, we can see that h = -3 and k = 7. So, the vertex of our parabola is at the point (-3, 7). This means the maximum value of the function is 7! Think about it: the squared term (x+3)^2 will always be greater than or equal to zero, no matter what value we plug in for x. Because of the negative sign in front, -(x+3)^2 will always be less than or equal to zero. So, the largest value f(x) can ever be is when -(x+3)^2 is zero, which happens when x = -3. At that point, f(x) = 0 + 7 = 7. This is why 7 is the maximum value.

Understanding the vertex is like finding the peak of a mountain range – it gives us the highest point the function reaches. From this, we can deduce the range, knowing that all other values of the function will be less than or equal to this maximum. By pinpointing the vertex, we're essentially defining the upper limit of our range, which is a critical step in solving the problem. It's like setting the boundary within which the function's output will always reside. This step solidifies our understanding of the function's behavior and brings us closer to the final answer.

Determining the Range

Alright, guys, we've nailed down the maximum value of the function. Now, let's use that knowledge to figure out the range. Remember, the range of a function is simply the set of all possible output values (the y-values) that the function can produce. Since our parabola opens downwards and has a maximum value of 7, the function can take on any value that is less than or equal to 7. It's like saying, "Okay, 7 is the highest we go, but we can go all the way down from there."

In mathematical notation, we express this range as f(x) ≤ 7. This means f(x) can be 7, or any number smaller than 7. We can visualize this on a number line, where the range would be represented by a closed circle at 7 (indicating that 7 is included) and an arrow extending to the left, showing that all numbers less than 7 are also included. This notation is a concise way of communicating the set of all possible y-values for our function.

So, in plain English, the range of the function f(x) = -(x+3)^2 + 7 is all real numbers less than or equal to 7. We've gone from understanding the function's structure to finding its maximum value, and now we've clearly defined the range. This entire process showcases the power of understanding the components of a function and how they influence its overall behavior. The range isn't just some abstract concept; it's a fundamental aspect of the function's identity, telling us what values it can produce. This understanding is key to tackling similar problems in the future and truly mastering the concepts of functions and their behavior.

Final Answer and Conclusion

So, there you have it! The range of the function f(x) = -(x+3)^2 + 7 is all real numbers less than or equal to 7. We arrived at this answer by carefully analyzing the function, identifying its key features (like the negative sign and the vertical shift), finding its maximum value, and then expressing the range in mathematical terms. This journey is a testament to the power of breaking down complex problems into smaller, more manageable steps.

By understanding the underlying principles of quadratic functions and their transformations, we were able to confidently determine the range. This is a valuable skill in mathematics, as it allows us to not just find answers but also understand the why behind them. Remember, math isn't just about memorizing formulas; it's about building a deep understanding of concepts and how they relate to each other. Keep practicing, keep exploring, and you'll become a math whiz in no time!

If you found this explanation helpful, give it a thumbs up and share it with your friends who might be struggling with similar problems. And don't forget to leave a comment below if you have any questions or want to explore other math topics. Keep learning, guys, and I'll see you in the next explanation!