graph y x2 2x 3

Guri Bee

2 min read

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20-10-2024

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When it comes to understanding quadratic functions, one of the most effective ways is to visualize them through graphs. In this article, we will delve into the graph of the quadratic function (y = x^2 + 2x + 3), and we'll cover key features such as the vertex, axis of symmetry, and intercepts. We'll also include practical examples to enhance comprehension.

Understanding the Function

The equation (y = x^2 + 2x + 3) is in the standard form of a quadratic function:

[ y = ax^2 + bx + c ]

In our case:

• (a = 1)
• (b = 2)
• (c = 3)

Since the coefficient (a) is positive, we know the parabola opens upwards.

Key Features of the Graph

1. Vertex: The vertex of a parabola can be found using the formula:

   [ x = -\frac{b}{2a} ]

   Plugging in our values:

   [ x = -\frac{2}{2 \times 1} = -1 ]

   Now we can find the corresponding (y)-value:

   [ y = (-1)^2 + 2(-1) + 3 = 1 - 2 + 3 = 2 ]

   So, the vertex is at the point ((-1, 2)).

2. Axis of Symmetry: The axis of symmetry for the graph is the vertical line that passes through the vertex, which can be expressed as:

   [ x = -1 ]

3. Y-Intercept: The y-intercept can be found by substituting (x = 0) into the function:

   [ y = 0^2 + 2(0) + 3 = 3 ]

   Therefore, the y-intercept is at the point ((0, 3)).

4. X-Intercepts: To find the x-intercepts, we set (y = 0) and solve the equation:

   [ 0 = x^2 + 2x + 3 ]

   Using the quadratic formula:

   [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

   Substituting our values:

   [ x = \frac{-2 \pm \sqrt{2^2 - 4(1)(3)}}{2(1)} = \frac{-2 \pm \sqrt{4 - 12}}{2} = \frac{-2 \pm \sqrt{-8}}{2} ]

   Since the discriminant ((b^2 - 4ac = -8)) is negative, there are no real x-intercepts, indicating that the graph does not cross the x-axis.

Graphical Representation

The graph of the function (y = x^2 + 2x + 3) would look like this:

   y
   |
  5|          *
  4|        *   *
  3|      *       *         (0,3)
  2|    *           *
  1|   *              *
   +------------------------ x
     -3  -2  -1  0  1

Analysis and Additional Explanations

The vertex of the parabola is its minimum point, which is a vital aspect for optimization problems in calculus. This function will always yield a value greater than or equal to 2, making it a useful function in scenarios where a minimum threshold is required.

Practical Example: Consider a scenario where a business analyzes its profits modeled by this quadratic equation. The minimum profit (vertex) occurs at the value of (x = -1), and the minimum profit is (y = 2). This can indicate that under certain conditions, the business will not incur losses; hence, it can strategize accordingly.

Conclusion

Understanding the graph of the quadratic function (y = x^2 + 2x + 3) gives insights not only into the function itself but also into various real-life applications where quadratic modeling is useful. Whether in business, physics, or engineering, grasping these concepts will enhance your analytical capabilities.

By recognizing the features of this graph, such as the vertex, axis of symmetry, and intercepts, you're now equipped to tackle quadratic functions with confidence! For further exploration, consider using graphing tools or software to visualize other quadratic equations and their attributes.

If you have any further questions or need additional explanations, feel free to reach out! Understanding quadratics opens doors to many mathematical concepts and real-world applications.