Find The Vertex Of F(x) = -x^2 + 4x - 3

When we're working with quadratic functions, one of the most important points to identify is the vertex. The vertex is essentially the highest or lowest point on the graph of the parabola, depending on whether the parabola opens upwards or downwards. For the function $f(x) = -x^2 + 4x - 3$, understanding its vertex is key to grasping its behavior. Let's dive deep into how we can find this crucial point. The vertex of a quadratic function in the standard form $f(x) = ax^2 + bx + c$ provides us with critical information about the function's maximum or minimum value and its axis of symmetry. The x-coordinate of the vertex can be found using the formula $x = -b / (2a)$. In our case, the function is $f(x) = -x^2 + 4x - 3$. Here, $a = -1$, $b = 4$, and $c = -3$. Plugging these values into the formula, we get $x = -4 / (2 * -1) = -4 / -2 = 2$. This tells us that the x-coordinate of our vertex is 2. To find the y-coordinate, we simply substitute this x-value back into our original function. So, $f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$. Therefore, the vertex of the function $f(x) = -x^2 + 4x - 3$ is at the point $(2, 1)$. This point represents the maximum value of the function because the coefficient 'a' is negative (-1), indicating that the parabola opens downwards.

Understanding Quadratic Functions and Their Vertices

Quadratic functions are a fundamental concept in algebra, characterized by their parabolic graphs. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants, and importantly, $a \neq 0$. The shape of this graph, a parabola, is determined by the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, meaning it has a minimum point. If 'a' is negative, the parabola opens downwards, and it has a maximum point. This specific maximum or minimum point is what we call the vertex. The vertex is not just any point; it's the turning point of the parabola. It lies on the axis of symmetry, a vertical line that divides the parabola into two mirror images. The equation of this axis of symmetry is always $x = -b / (2a)$, which is conveniently the same as the x-coordinate of the vertex itself. Understanding the vertex helps us sketch the graph accurately, determine the range of the function, and solve various optimization problems in mathematics and science. For instance, in physics, the trajectory of a projectile can often be modeled by a quadratic function, and its vertex would represent the highest point the projectile reaches. In economics, the cost or profit functions might be quadratic, and the vertex could indicate the minimum cost or maximum profit. Therefore, mastering the skill of finding the vertex is a significant step in comprehending quadratic functions and their real-world applications. Let's reiterate the process for our specific function $f(x) = -x^2 + 4x - 3$. We identified $a = -1$, $b = 4$, and $c = -3$. The x-coordinate of the vertex is calculated using $x = -b / (2a)$, which yields $x = -4 / (2 * -1) = -4 / -2 = 2$. Once we have the x-coordinate, we substitute it back into the function to find the corresponding y-coordinate: $f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$. So, the vertex is indeed $(2, 1)$. This point is crucial as it signifies the peak of the downward-opening parabola, representing the maximum output of the function.

Methods for Finding the Vertex

There are several reliable methods to find the vertex of a quadratic function. The most common and often the most straightforward is using the vertex formula derived from completing the square or calculus. As we've already discussed, for a quadratic function in the standard form $f(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by $x = -b / (2a)$. Once this x-value is found, you substitute it back into the function $f(x)$ to calculate the corresponding y-coordinate. This method is efficient and directly applicable to any quadratic function presented in standard form. Another powerful technique is completing the square. This method transforms the standard form of the quadratic equation into vertex form, $f(x) = a(x-h)^2 + k$, where $(h, k)$ are the coordinates of the vertex. Let's apply this to $f(x) = -x^2 + 4x - 3$. First, we can factor out the coefficient 'a' from the terms involving x: $f(x) = -1(x^2 - 4x) - 3$. Now, we focus on the expression inside the parentheses, $x^2 - 4x$. To complete the square, we take half of the coefficient of the x term (-4), square it ($(-4/2)^2 = (-2)^2 = 4$), and add and subtract it inside the parentheses: $f(x) = -(x^2 - 4x + 4 - 4) - 3$. Now, we can rewrite the perfect square trinomial $x^2 - 4x + 4$ as $(x - 2)^2$: $f(x) = -((x - 2)^2 - 4) - 3$. Distributing the negative sign, we get $f(x) = -(x - 2)^2 + 4 - 3$. Simplifying, we arrive at the vertex form: $f(x) = -(x - 2)^2 + 1$. Comparing this to the general vertex form $f(x) = a(x-h)^2 + k$, we can see that $h = 2$ and $k = 1$. Thus, the vertex is $(h, k) = (2, 1)$. This method not only gives us the vertex but also reveals the axis of symmetry ($x=h$) and the vertical stretch/compression and reflection. A third approach, particularly useful if you are familiar with calculus, is to use derivatives. The vertex of a parabola is where the slope of the tangent line is zero (since it's a horizontal tangent at the maximum or minimum). So, we can find the derivative of $f(x)$: $f'(x) = d/dx (-x^2 + 4x - 3) = -2x + 4$. Setting the derivative equal to zero to find critical points: $-2x + 4 = 0$. Solving for x, we get $-2x = -4$, which means $x = 2$. This x-value is the x-coordinate of the vertex. Substituting $x = 2$ back into the original function $f(x)$ gives us the y-coordinate: $f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$. So, again, the vertex is $(2, 1)$. Each of these methods provides a robust way to locate the vertex, and choosing the best one often depends on personal preference or the specific form in which the quadratic function is presented.

Analyzing the Options Provided

We've established that the vertex of the function $f(x) = -x^2 + 4x - 3$ is at the point $(2, 1)$. Now, let's look at the options provided to see which one matches our result. The options are:

A) $(-2,-13)$ B) $(2,-13)$ C) $(2,1)$ D) $(2,-1)$

By comparing our calculated vertex $(2, 1)$ with these options, we can clearly see that Option C) $(2,1)$ is the correct answer. It's important to be meticulous when performing calculations. A small arithmetic error can lead to choosing an incorrect option. For example, if we had mistakenly calculated the y-coordinate and arrived at -1, we might have been tempted to choose option D. Or, if we had made an error with the sign of 'a' or 'b' when calculating the x-coordinate, we might have ended up with -2, leading us to options A or B. Let's quickly re-verify our steps to ensure accuracy. Using the formula $x = -b / (2a)$: $x = -4 / (2 * -1) = -4 / -2 = 2$. This is correct. Now, substituting $x=2$ into $f(x)$: $f(2) = -(2)^2 + 4(2) - 3 = -4 + 8 - 3 = 1$. This is also correct. The vertex is indeed $(2,1)$.

It's also worth noting why the other options are incorrect. For option A, $(-2, -13)$: if $x = -2$, then $f(-2) = -(-2)^2 + 4(-2) - 3 = -(4) - 8 - 3 = -4 - 8 - 3 = -15$. So, $(-2, -15)$ is a point on the graph, not the vertex. For option B, $(2, -13)$: we already calculated that when $x=2$, $f(2)=1$. So $(2, -13)$ is incorrect. For option D, $(2, -1)$: again, when $x=2$, $f(2)=1$, not -1. Therefore, only option C correctly represents the vertex of the given quadratic function. This rigorous checking process confirms our finding and reinforces the understanding that precise calculation is paramount when dealing with mathematical problems.

Conclusion: The Vertex is $(2, 1)$

In summary, we've thoroughly explored how to find the vertex of the quadratic function $f(x) = -x^2 + 4x - 3$. We employed the standard formula $x = -b / (2a)$ to find the x-coordinate, which resulted in $x = 2$. Subsequently, we substituted this value back into the function to determine the y-coordinate, yielding $y = 1$. Thus, the vertex is located at the point $(2, 1)$. We also touched upon alternative methods like completing the square and using calculus, all of which corroborate our finding. Recognizing the vertex is fundamental to understanding the behavior and graphing of quadratic functions, as it signifies the maximum or minimum point of the parabola. For $f(x) = -x^2 + 4x - 3$, the negative coefficient of the $x^2$ term indicates that the parabola opens downwards, making $(2, 1)$ the maximum point. This understanding is invaluable for solving a wide array of mathematical and scientific problems, from analyzing projectile motion to optimizing economic models. Always double-check your calculations to avoid errors, as demonstrated by our analysis of the provided options.

For further exploration into quadratic functions and their properties, you can visit reliable resources such as Khan Academy's section on quadratic functions or consult the detailed explanations on Wolfram MathWorld regarding parabolas.