Vertex Coordinates Of Y = X² + 2x + 1: Find It Now!

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Vertex Coordinates of y = x² + 2x + 1: Find it Now!

Hey guys! Today, we're diving into a super important concept in math: finding the vertex of a quadratic function. Specifically, we're going to figure out the vertex coordinates for the quadratic function y = x² + 2x + 1. This is a classic problem, and once you understand the method, you'll be able to solve similar problems in a snap. So, let's get started!

Understanding Quadratic Functions

First, let's quickly recap what a quadratic function is. A quadratic function is a polynomial function of degree two, generally written in 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, which is a U-shaped curve. The vertex of the parabola is the point where the parabola changes direction. If a > 0, the parabola opens upwards, and the vertex is the minimum point. If a < 0, the parabola opens downwards, and the vertex is the maximum point.

In our case, the quadratic function is y = x² + 2x + 1. Here, a = 1, b = 2, and c = 1. Since a is positive (1 > 0), the parabola opens upwards, and the vertex will be the minimum point.

Methods to Find the Vertex

There are a couple of ways to find the vertex of a quadratic function. Let's explore the two most common methods:

1. Using the Vertex Formula

The vertex formula is a straightforward way to find the coordinates of the vertex. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex (h) is given by:

h = -b / 2a

Once you find the x-coordinate (h), you can find the y-coordinate (k) by plugging h back into the original function:

k = f(h)

Let's apply this to our function y = x² + 2x + 1:

a = 1, b = 2, c = 1

h = -2 / (2 * 1) = -2 / 2 = -1

Now, we find the y-coordinate (k) by plugging h = -1 into the function:

k = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0

So, the vertex coordinates are (-1, 0).

2. Completing the Square

Completing the square is another powerful method to find the vertex. This method involves rewriting the quadratic function in vertex form, which is:

f(x) = a(x - h)² + k

Where (h, k) is the vertex of the parabola. Let's apply this to our function y = x² + 2x + 1:

y = x² + 2x + 1

Notice that the given quadratic function is already a perfect square trinomial. It can be factored as:

y = (x + 1)²

This can be rewritten as:

y = 1(x - (-1))² + 0

Comparing this with the vertex form f(x) = a(x - h)² + k, we can see that:

h = -1

k = 0

So, the vertex coordinates are (-1, 0). This method confirms our result from the vertex formula.

Step-by-Step Solution

Let's break down the solution into clear, easy-to-follow steps:

  1. Identify the coefficients: In the quadratic function y = x² + 2x + 1, identify the coefficients a, b, and c. Here, a = 1, b = 2, and c = 1.
  2. Use the vertex formula: The x-coordinate of the vertex (h) is given by h = -b / 2a. Plug in the values of a and b: h = -2 / (2 * 1) = -1
  3. Find the y-coordinate: Plug the x-coordinate h = -1 into the original function to find the y-coordinate (k): k = (-1)² + 2(-1) + 1 = 1 - 2 + 1 = 0
  4. State the vertex coordinates: The vertex coordinates are (-1, 0).

Analyzing the Options

Now that we've found the vertex coordinates, let's analyze the given options:

A) (-1, 0) B) (0, 1) C) (-2, -1) D) (1, 2)

Our calculated vertex coordinates are (-1, 0), which matches option A.

Why the Other Options Are Incorrect

It's essential to understand why the other options are incorrect to solidify our understanding.

  • Option B (0, 1): This point is the y-intercept of the function, where the parabola crosses the y-axis. It's not the vertex.
  • Option C (-2, -1): This point does not satisfy the vertex formula, nor does it lie on the axis of symmetry. Plugging x = -2 into the function gives y = (-2)² + 2(-2) + 1 = 4 - 4 + 1 = 1, not -1.
  • Option D (1, 2): This point is also not the vertex. Plugging x = 1 into the function gives y = (1)² + 2(1) + 1 = 1 + 2 + 1 = 4, not 2.

Final Answer

Therefore, the correct answer is A) (-1, 0). The vertex of the quadratic function y = x² + 2x + 1 is located at the point (-1, 0). This represents the minimum value of the function, as the parabola opens upwards.

Tips and Tricks

  • Memorize the vertex formula: Knowing the vertex formula h = -b / 2a can save you time on tests and assignments.
  • Practice completing the square: Completing the square is a useful technique for various quadratic function problems, not just finding the vertex.
  • Visualize the parabola: Understanding that the vertex is either the minimum or maximum point can help you intuitively check your answer.
  • Double-check your calculations: Always double-check your calculations to avoid simple arithmetic errors.

Conclusion

In conclusion, finding the vertex of a quadratic function is a fundamental skill in algebra. By using the vertex formula or completing the square, we can easily determine the vertex coordinates. Remember to understand the concept and practice regularly to master it. I hope this guide has been helpful, and you now feel confident in tackling similar problems. Keep practicing, and you'll become a pro at finding vertices in no time! You got this, guys!