Find The Vertex Of A Parabola: $f(x)=3(x-3)^2+4$

Understanding the Vertex Form of a Parabola

When we talk about parabolas, one of the most crucial pieces of information we often want to find is its vertex. The vertex is essentially the turning point of the parabola – it's either the minimum point (if the parabola opens upwards) or the maximum point (if it opens downwards). Understanding the vertex is key to graphing parabolas accurately and solving various problems involving quadratic functions. Fortunately, when a quadratic function is presented in a specific format, often called the vertex form, identifying the vertex becomes incredibly straightforward. The standard vertex form of a quadratic equation is written as $f(x) = a(x-h)^2 + k$, where $(h, k)$ directly represents the coordinates of the vertex. In this form, 'a' dictates the direction and width of the parabola, while 'h' and 'k' pinpoint the exact location of the vertex on the coordinate plane. The value of 'h' tells us the horizontal shift of the parabola from its parent function $y=x^2$, and 'k' tells us the vertical shift. This form is incredibly powerful because it bypasses the need for complex calculations like completing the square or using the $-b/2a$ formula when the equation is already in this shape. It’s like having a cheat code for finding the vertex! So, if you see a quadratic equation in the form $f(x) = a(x-h)^2 + k$, take a moment to appreciate how easily you can extract the vertex coordinates just by looking at the numbers represented by 'h' and 'k'. Remember, the sign of 'h' is important; it’s $(x-h)$, so if you see $(x+5)$, then $h$ is actually $-5$. This might seem like a small detail, but it's a common place where mistakes can happen. Keep this vertex form in mind as we move forward, as it’s the key to unlocking the solution for our specific problem.

Identifying the Vertex in $f(x)=3(x-3)^2+4$

Now that we've got a solid grasp on the vertex form, let's apply that knowledge to our specific equation: $f(x)=3(x-3)^2+4$. Our goal is to find the vertex, and we know that the vertex form is $f(x) = a(x-h)^2 + k$. Comparing our given equation to the standard vertex form, we can directly identify the values of 'a', 'h', and 'k'. Let's break it down: The coefficient 'a' is the number multiplying the squared term. In our case, $a = 3$. This 'a' value tells us that the parabola opens upwards (because it's positive) and is narrower than the standard $y=x^2$ parabola. The 'h' value is found within the parentheses, associated with the 'x' term. We have $(x-3)^2$. In the standard form, it's $(x-h)^2$. By direct comparison, we can see that $h = 3$. It's crucial to pay attention to the sign here. Since our equation has $(x-3)$, our 'h' is positive 3. If it were $(x+3)$, then 'h' would be -3. Finally, the 'k' value is the constant term added outside the squared expression. In our equation, we have $+4$. Comparing this to $+k$ in the standard form, we can confidently say that $k = 4$. So, by simply comparing $f(x)=3(x-3)^2+4$ with the general vertex form $f(x) = a(x-h)^2 + k$, we've successfully identified that $a=3$, $h=3$, and $k=4$. This is the beauty of the vertex form – it makes identifying these key components almost instantaneous. The vertex of the parabola is represented by the ordered pair $(h, k)$. Therefore, with $h=3$ and $k=4$, the vertex of the parabola $f(x)=3(x-3)^2+4$ is $(3, 4)$. This ordered pair pinpoints the exact location where the parabola changes direction. It’s the minimum point of this particular parabola because the 'a' value (3) is positive, meaning the parabola opens upwards.

Why is the Vertex Important?

The vertex of a parabola holds significant importance in understanding its behavior and properties. It's not just a random point; it's the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards, respectively. This maximum or minimum value is often referred to as the extreme value of the quadratic function. For instance, in real-world applications, if a quadratic function models the trajectory of a projectile, the vertex would represent the peak height reached by the projectile. Similarly, if a function describes the profit of a company based on production levels, the vertex could indicate the maximum profit or the minimum loss. The vertex also plays a crucial role in graphing parabolas. Knowing the vertex allows us to accurately sketch the curve, as it serves as a central anchor point. From the vertex, we can determine the axis of symmetry, which is a vertical line passing through the vertex. This line divides the parabola into two mirror images, making it easier to plot additional points and complete the graph. The equation of the axis of symmetry for a parabola in vertex form $f(x) = a(x-h)^2 + k$ is simply $x=h$. In our specific case, $f(x)=3(x-3)^2+4$, the vertex is $(3, 4)$, so the axis of symmetry is the line $x=3$. This means the parabola is symmetric around this vertical line. Furthermore, the vertex helps us understand the range of the quadratic function. If the parabola opens upwards (a > 0), the vertex represents the minimum y-value, and the range is $[k, o)$. If it opens downwards (a < 0), the vertex represents the maximum y-value, and the range is $(- o, k]$. For $f(x)=3(x-3)^2+4$, since $a=3$ is positive, the vertex $(3, 4)$ is the minimum point, and the range of the function is $[4, o)$. Understanding the vertex is fundamental for solving optimization problems, analyzing data that follows a parabolic pattern, and gaining a comprehensive understanding of quadratic functions. It provides a snapshot of the function's extreme behavior and its position on the coordinate plane.

Conclusion

In summary, the vertex of a parabola is a pivotal point that reveals its highest or lowest value and dictates its axis of symmetry. For the given quadratic function, $f(x)=3(x-3)^2+4$, we've successfully determined its vertex by recognizing that the equation is already presented in the vertex form, $f(x) = a(x-h)^2 + k$. By directly comparing the two forms, we identified $h=3$ and $k=4$. Therefore, the vertex of the parabola is the ordered pair (3, 4). This point signifies the turning point of the parabola, and since the leading coefficient (3) is positive, it represents the minimum value of the function. This straightforward method highlights the efficiency of using the vertex form for analyzing parabolas. For further exploration into parabolas and quadratic functions, you can consult resources like Khan Academy's section on quadratic functions, which offers detailed explanations and practice exercises. You might also find the Wolfram MathWorld entry on parabolas to be an invaluable academic resource for deeper mathematical insights.