Vertex Of Quadratic Function: F(x) = -x^2 + 8x - 14
Hey guys! Let's dive into how to find the vertex of a quadratic function. We'll break down the equation f(x) = -x² + 8x - 14 step by step, so you'll nail this concept in no time. Understanding the vertex is super important in math because it tells us the highest or lowest point of the parabola, which can be useful in many real-world applications. Think of things like the trajectory of a ball or optimizing business costs – the vertex helps us find those key maximum or minimum points. Let's get started and make sure you understand this crucial concept!
Understanding Quadratic Functions
Before we jump into finding the vertex, let's make sure we're all on the same page about quadratic functions. A quadratic function is a polynomial function of the form:
f(x) = ax² + bx + c
where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve. This parabola can open upwards (if a > 0) or downwards (if a < 0). The vertex is the point where the parabola changes direction – it’s either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards).
In our example, f(x) = -x² + 8x - 14, we have:
- a = -1
- b = 8
- c = -14
Since 'a' is negative (-1), the parabola opens downwards, meaning the vertex will be the highest point on the curve. Knowing this helps us anticipate the nature of the vertex we're about to find.
Methods to Find the Vertex
There are a couple of ways we can find the vertex of a quadratic function. We'll go through two common methods:
- Using the Vertex Formula
- Completing the Square
Let's tackle each method one by one to give you a solid understanding.
1. Using the Vertex Formula
The vertex formula is a straightforward way to find the coordinates of the vertex. The vertex of a quadratic function in the form f(x) = ax² + bx + c is given by the point (h, k), where:
- h = -b / 2a
- k = f(h) (This means we plug the value of 'h' back into the original function to find 'k')
This formula is super handy because it directly gives us the x-coordinate ('h') of the vertex, and then we just plug that value back into the function to get the y-coordinate ('k'). Let's apply this to our example function, f(x) = -x² + 8x - 14.
Applying the Vertex Formula to f(x) = -x² + 8x - 14
First, identify 'a' and 'b' from our function:
- a = -1
- b = 8
Now, let's calculate 'h':
h = -b / 2a = -8 / (2 * -1) = -8 / -2 = 4
So, the x-coordinate of the vertex is 4. Now we need to find the y-coordinate, 'k'. We do this by plugging 'h' (which is 4) back into the original function:
k = f(4) = -(4)² + 8(4) - 14
Let's simplify:
k = -16 + 32 - 14 = 2
Therefore, the vertex of the quadratic function f(x) = -x² + 8x - 14 is (4, 2). The vertex formula is a reliable method, especially when you need a quick and direct way to find the vertex.
2. Completing the Square
Another method for finding the vertex is by completing the square. This method involves rewriting the quadratic function in vertex form. The vertex form of a quadratic function is:
f(x) = a(x - h)² + k
where (h, k) is the vertex of the parabola. Completing the square might seem a bit tricky at first, but it's a powerful technique that not only helps find the vertex but also provides a deeper understanding of quadratic functions.
Steps to Complete the Square
Let's go through the steps to complete the square for our function, f(x) = -x² + 8x - 14:
-
Factor out 'a' from the x² and x terms: In our case, a = -1, so we factor out -1 from the first two terms:
f(x) = -1(x² - 8x) - 14
-
Complete the square inside the parentheses: To complete the square, we need to add and subtract the square of half the coefficient of the x term. The coefficient of our x term is -8. Half of -8 is -4, and (-4)² is 16. So, we add and subtract 16 inside the parentheses:
f(x) = -1(x² - 8x + 16 - 16) - 14
-
Rewrite the trinomial as a perfect square: The expression x² - 8x + 16 is a perfect square and can be written as (x - 4)²:
f(x) = -1((x - 4)² - 16) - 14
-
Distribute 'a' and simplify: Distribute the -1 back into the parentheses:
f(x) = -(x - 4)² + 16 - 14
-
Combine constants: Combine the constant terms:
f(x) = -(x - 4)² + 2
Now our function is in vertex form: f(x) = -(x - 4)² + 2. Comparing this to the vertex form f(x) = a(x - h)² + k, we can see that:
- h = 4
- k = 2
So, the vertex is (4, 2), which matches the result we got using the vertex formula!
Comparing the Methods
Both the vertex formula and completing the square are effective methods for finding the vertex of a quadratic function, but they have slightly different strengths:
- Vertex Formula: This method is quick and direct, making it great for exams or situations where time is limited. You just plug in the values and calculate.
- Completing the Square: This method is a bit more involved, but it's incredibly valuable for understanding the structure of quadratic functions. It rewrites the function in vertex form, which directly shows the vertex and the shape of the parabola. Completing the square is also useful in other areas of math, like calculus.
The best method to use often depends on the context and your personal preference. If you just need the vertex quickly, the formula is your best bet. If you want a deeper understanding and might need to manipulate the function further, completing the square is a fantastic tool.
Conclusion
Alright, guys, we've covered how to find the vertex of the quadratic function f(x) = -x² + 8x - 14 using both the vertex formula and completing the square. We found that the vertex is (4, 2). Remember, the vertex is a crucial point on the parabola, representing its maximum or minimum value, and understanding how to find it is a key skill in algebra.
Whether you prefer the direct approach of the vertex formula or the comprehensive understanding gained from completing the square, you now have two solid methods in your toolkit. Keep practicing, and you’ll become a pro at finding vertices in no time!
If you have any questions or want to dive deeper into quadratic functions, feel free to ask. Happy math-ing!
