Finding The Decreasing Interval: A Quadratic Function Deep Dive

Hey everyone! Let's dive into a classic math problem that often pops up: determining the interval where a quadratic function is decreasing. Specifically, we're looking at the function $f(x) = 2(x + 3)^2 + 2$. Don't worry, it might seem a bit intimidating at first, but trust me, we'll break it down step-by-step to make it super clear. This is a fundamental concept in understanding the behavior of quadratic functions, which are incredibly useful in all sorts of real-world applications. Let's get started, shall we?

Understanding Quadratic Functions and Their Graphs

First off, let's refresh our memories about quadratic functions. A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants, and crucially, a is not equal to zero. The graph of a quadratic function is a parabola – a U-shaped curve. This curve can either open upwards (if a > 0) or downwards (if a < 0). In our case, the given function is $f(x) = 2(x + 3)^2 + 2$. This function is in vertex form, which is super convenient for us. The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where (h, k) represents the vertex of the parabola. Since the coefficient of the squared term (which is 2 in our case) is positive, our parabola opens upwards. This means it has a minimum point, and it's decreasing on the left side of that minimum point. To find the interval where the graph is decreasing, we really need to pinpoint where that vertex sits. Knowing the vertex is key to understanding where the function changes direction, from decreasing to increasing (or vice-versa, if the parabola opened downwards).

Let's get even more specific. The vertex form is awesome because it directly gives us the vertex coordinates. In the form $f(x) = a(x - h)^2 + k$, the vertex is at the point (h, k). Looking at our function, $f(x) = 2(x + 3)^2 + 2$, we can see that it's already in vertex form. We can rewrite it as $f(x) = 2(x - (-3))^2 + 2$. Therefore, the vertex is at the point (-3, 2). This is the lowest point on our parabola. Since the parabola opens upwards, it decreases as we move from left to right up to the x-coordinate of the vertex and then starts increasing. So, the interval where the graph is decreasing will be all the x-values to the left of the vertex's x-coordinate. To visualize this, picture the parabola. It starts high on the left, goes down to the vertex at (-3, 2), and then goes up on the right. Therefore, the function is decreasing from negative infinity up to -3. Got it? Let's solidify our understanding with some more details.

Identifying the Vertex and the Axis of Symmetry

Okay, guys, let's talk about the vertex and the axis of symmetry, which are besties when it comes to understanding parabolas. As we saw, our function, $f(x) = 2(x + 3)^2 + 2$, is in vertex form. This is super helpful because it immediately tells us the vertex. The vertex form is $f(x) = a(x - h)^2 + k$, where the vertex is at the point (h, k). In our case, $f(x) = 2(x + 3)^2 + 2$, which we can rewrite as $f(x) = 2(x - (-3))^2 + 2$. So, the vertex is at (-3, 2). Now, the axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two symmetrical halves. The equation of the axis of symmetry is always $x = h$, where h is the x-coordinate of the vertex. For our function, the axis of symmetry is $x = -3$. This axis is crucial because it's the dividing line where the function changes direction. To the left of the axis of symmetry, the function is decreasing (because our parabola opens upwards). To the right of the axis of symmetry, the function is increasing. The x-coordinate of the vertex is where the change happens. Now, let's get back to the core concept of this problem. Remember, we are looking for the interval where the function is decreasing. Since our parabola opens upwards and has a vertex at (-3, 2), the function decreases from negative infinity up to the x-coordinate of the vertex, which is -3. This means that the graph of the function is decreasing over the interval $(-\infty, -3)$.

Think about it like this: as you move from left to right along the graph, the function values (the y-values) are getting smaller until you reach x = -3. After that point, the function values start to increase. This is why understanding the vertex is so fundamental; it’s the key to unlocking the behavior of the entire quadratic function. We're getting closer to solving this problem, but let's go a bit deeper to make sure we've covered all the bases.

Determining the Decreasing Interval

Alright, let's put it all together to pinpoint the interval where our function, $f(x) = 2(x + 3)^2 + 2$, is decreasing. We’ve already done a lot of the heavy lifting. We know our parabola opens upwards because the coefficient of the squared term (2) is positive. We've also figured out that the vertex of the parabola is at (-3, 2). Because the parabola opens upwards, it has a minimum point, which is the vertex. To the left of the vertex, the function is decreasing; to the right of the vertex, the function is increasing. The decreasing interval is, therefore, everything to the left of the x-coordinate of the vertex. Since the x-coordinate of the vertex is -3, the function is decreasing over the interval $(-\infty, -3)$. This means that for any x-value less than -3, the value of the function is getting smaller as we move along the graph. Now, let's check the answer choices provided in the original question to make sure we've got the correct one. The choices were: A. $(-\infty, -3)$ B. $(-\infty, 2)$ C. $(-3, \infty)$ D. $(2, \infty)$.

Looking at these, we see that option A, $(-\infty, -3)$, matches perfectly with our finding. The function is decreasing from negative infinity up to, but not including, -3. The other options don't make sense: Option B refers to the y-coordinate of the vertex, not the x-coordinate that determines the direction of change. Option C refers to where the function is increasing, and Option D is incorrect for similar reasons. So, the correct answer is A. And there you have it, guys! We've successfully navigated the problem, understanding the behavior of the quadratic function by leveraging the vertex form and the properties of parabolas. Remember, understanding the vertex and the direction the parabola opens are crucial steps to solving these types of problems. Let me know if you have any questions. We can also explore more examples.

Conclusion: Mastering Quadratic Functions

In conclusion, we've successfully tackled the question of finding the decreasing interval for the quadratic function $f(x) = 2(x + 3)^2 + 2$. We've walked through the key concepts: understanding the nature of quadratic functions, recognizing the vertex form, identifying the vertex, and using the vertex to determine the intervals of increase and decrease. The graph is decreasing over the interval $(-\infty, -3)$. This approach isn't just about getting the right answer; it's about developing a deeper understanding of quadratic functions. Quadratic functions pop up all the time in mathematics and real-world scenarios. This is why knowing how to analyze these functions is a super valuable skill. Think about projectile motion, the shape of satellite dishes, or even the trajectory of a basketball – all described by quadratic functions. By understanding concepts like the vertex, axis of symmetry, and the direction of the parabola's opening, you gain powerful tools for problem-solving. Practice with different examples will solidify this knowledge. You can change the coefficients and constants in a quadratic function to see how the graph changes, which helps build intuition. Keep practicing, keep exploring, and you'll find yourselves acing these kinds of problems with ease. If you found this explanation helpful, please let me know. Thanks for hanging out, and keep learning, everyone! Remember, the more problems you solve, the more confident you'll become. So, keep up the great work, and good luck with your future math endeavors!