Understanding Quadratic Functions: Analyzing Graphs

Hey math enthusiasts! Let's dive into the fascinating world of quadratic functions and their graphs. Today, we're going to break down the quadratic function $f(x) = 5(x - 3)^2 + 2$ and figure out what its graph looks like. This function is presented in vertex form, which gives us some awesome clues about the graph's behavior. We'll explore key features like whether the graph opens upward or downward, and where its maximum or minimum value can be found. By the end of this, you'll be pros at understanding and analyzing quadratic functions. Ready to get started, guys? Let's go!

Unveiling the Secrets of Quadratic Functions

First off, let's talk about what a quadratic function is. Simply put, it's a function that can be written in the form $f(x) = ax^2 + bx + c$, where a, b, and c are constants, and $a$ is not equal to zero. These functions are super important in math and have all sorts of real-world applications – from physics and engineering to economics and even sports! The graph of a quadratic function always forms a special U-shaped curve called a parabola. The neat thing about parabolas is that they're symmetrical, meaning they have a line of symmetry that cuts the curve exactly in half. Now, when dealing with the graph of a quadratic function, there are two key features to look at: whether it opens upward or downward and where the vertex is located. The vertex is the most important point on the parabola. It's either the lowest point (the minimum value) if the parabola opens upward or the highest point (the maximum value) if it opens downward. The 'a' value in the standard quadratic equation $f(x) = ax^2 + bx + c$ dictates the direction the parabola opens. If a is positive, the parabola opens upwards (like a smile!), and if a is negative, it opens downwards (like a frown!).

Let’s apply this to our function, $f(x) = 5(x - 3)^2 + 2$. Notice that the 'a' value here is 5, which is a positive number. That tells us right away that the parabola opens upwards. This means that the graph has a minimum value and no maximum value because it extends upwards infinitely. The vertex is an important point to understand when analyzing the graph. Now, let's determine the coordinates of the vertex, which will further help us understand the graph and answer the initial question. This is the heart of the matter when it comes to understanding quadratic functions and their graphs. Understanding the basics will make solving problems much easier. The form $f(x) = a(x - h)^2 + k$ is called the vertex form of a quadratic function. It’s incredibly useful because it makes it super easy to identify the vertex, which is located at the point (h, k). In our function, $f(x) = 5(x - 3)^2 + 2$, we can see that h = 3 and k = 2. Therefore, the vertex of the parabola is at the point (3, 2). This also means that the parabola has a minimum value at the point (3, 2) since it opens upward.

Detailed Analysis of the Function $f(x) = 5(x - 3)^2 + 2$

Now, let's thoroughly analyze our function to ensure that we understand every aspect of its graph. As we've already discussed, the function is given in vertex form: $f(x) = 5(x - 3)^2 + 2$. The coefficient of the squared term (which is 5 in this case) tells us the direction of the parabola. Since 5 is a positive number, the parabola opens upwards. This means the graph has a minimum point but no maximum point. The vertex form provides us with the coordinates of the vertex. The vertex form is specifically designed to make it easy to identify the vertex directly. Recall that the vertex form is $f(x) = a(x - h)^2 + k$, and the vertex is at the point (h, k). In our function, the vertex is at (3, 2). This is a crucial piece of information. The vertex is the point at which the parabola changes direction. Since the parabola opens upward, the vertex (3, 2) represents the minimum point of the graph. The y-coordinate of the vertex (which is 2 in this case) is the minimum value of the function. Understanding the vertex is key to answering questions about quadratic functions and their graphs, and it unlocks a deeper understanding of the function's behavior. We can see that the line of symmetry is at the vertical line $x = 3$. This means that the parabola is symmetrical around this line. The points on the parabola are equidistant from this line. This is a very interesting property of parabolas and quadratic functions.

Now, let’s revisit the multiple-choice options we started with.

• A. Opens upward and has maximum value at (3, 2): This is incorrect because the parabola opens upwards, but it has a minimum value at the vertex, not a maximum value.
• B. Opens upward and has minimum value at (3, 2): This is correct! The parabola opens upwards (because the coefficient of the squared term is positive), and its minimum value is at the vertex (3, 2).
• C. Opens downward and has minimum value at (3, 2): This is incorrect. The parabola opens downward if the coefficient of the squared term is negative, and in this case, the minimum value would be a maximum value. However, the graph opens upward.
• D. Opens downward and has maximum value at (3, 2): This is incorrect. The parabola opens downwards if the coefficient of the squared term is negative, but the graph opens upwards in our example.

So, the correct answer is B! Understanding each component of the quadratic function allows you to identify key details about the graph.

Key Takeaways and Conclusion

Alright, guys, let's sum up what we've learned! When dealing with a quadratic function, you should focus on these things:

• The direction the parabola opens: This depends on the sign of the coefficient of the $x^2$ term (or the 'a' value in $f(x) = ax^2 + bx + c$). Positive means upwards, negative means downwards.
• The vertex: This is the most important point on the parabola. It's the minimum point if the parabola opens upwards and the maximum point if it opens downwards. If the function is in vertex form, finding the vertex is easy!

By knowing these two key pieces of information, you can accurately describe and sketch the graph of any quadratic function. Analyzing these types of problems is much easier when you break it down into smaller parts. Keep practicing, and you'll be a quadratic function whiz in no time. I hope this helps you understand quadratic functions better. Keep practicing, and you'll get the hang of it! See ya in the next lesson!