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Exploring the Graph of y = 2x² + 1: A Comprehensive Guide

The equation y = 2x² + 1 represents a parabola, a fundamental curve in mathematics with widespread applications in various fields, from physics and engineering to economics and computer graphics. Understanding its graph involves analyzing its key features: vertex, axis of symmetry, concavity, and intercepts. This comprehensive guide will delve into each aspect, providing a detailed understanding of the parabola represented by y = 2x² + 1.

Understanding the Parabola's Form

The equation y = 2x² + 1 is a quadratic equation in the standard form y = ax² + bx + c, where a = 2, b = 0, and c = 1. The coefficient 'a' dictates the parabola's concavity and vertical stretch or compression. A positive 'a' (as in our case) indicates that the parabola opens upwards, forming a U-shape. The absolute value of 'a' determines the parabola's vertical scaling; a value greater than 1, like our 'a' = 2, implies a vertical stretch, making the parabola narrower than the basic parabola y = x². The 'c' value (1 in this case) represents the y-intercept, indicating where the parabola crosses the y-axis.

Key Features of y = 2x² + 1

• Vertex: The vertex is the lowest (or highest, for downward-opening parabolas) point on the parabola. For a parabola in the form y = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / 2a. In our case, with b = 0 and a = 2, the x-coordinate is 0. Substituting x = 0 into the equation gives y = 1. Therefore, the vertex is at (0, 1).

• Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It always passes through the vertex. For a parabola in the form y = ax² + bx + c, the axis of symmetry is given by the equation x = -b / 2a. In this case, the axis of symmetry is x = 0, which is the y-axis.

• Concavity: As mentioned earlier, the positive value of 'a' (a = 2) indicates that the parabola opens upwards, meaning it is concave up.

• Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the equation gives y = 1. Therefore, the y-intercept is (0, 1). Note that in this specific case, the y-intercept coincides with the vertex.

• X-intercepts (Roots): The x-intercepts are the points where the parabola intersects the x-axis. These are found by setting y = 0 and solving for x. In our case:

  0 = 2x² + 1 2x² = -1 x² = -1/2 x = ±√(-1/2)

Since the square root of a negative number is imaginary, this parabola has no real x-intercepts. It does not cross the x-axis.

Graphing the Parabola

Now that we've identified the key features, let's sketch the graph:

1. Plot the vertex: Mark the point (0, 1) on the coordinate plane.

2. Draw the axis of symmetry: Draw a vertical line through the vertex (x = 0).

3. Plot additional points: Since the parabola is symmetrical, we only need to find points on one side of the axis of symmetry and then mirror them. Let's find the y-values for x = 1 and x = -1:

   • If x = 1, y = 2(1)² + 1 = 3. So we have the point (1, 3).
   • If x = -1, y = 2(-1)² + 1 = 3. So we have the point (-1, 3).

4. Plot the points and sketch the curve: Plot the points (1, 3) and (-1, 3), and their reflections across the axis of symmetry. Draw a smooth, U-shaped curve through these points, ensuring it passes through the vertex and opens upwards. Remember that the parabola extends infinitely in both directions.

Transformations and Comparisons

Understanding the graph of y = 2x² + 1 can be enhanced by comparing it to the basic parabola y = x². The equation y = 2x² + 1 represents a transformation of the basic parabola:

• Vertical Stretch: The coefficient 2 stretches the parabola vertically, making it narrower than y = x².

• Vertical Shift: The constant term +1 shifts the parabola one unit upward.

By comparing it with y = x², we can visually appreciate the effects of these transformations on the shape and position of the parabola.

Applications of Parabolas

Parabolas, represented by quadratic equations like y = 2x² + 1, are not just abstract mathematical concepts; they have numerous practical applications:

• Physics: The trajectory of a projectile under the influence of gravity follows a parabolic path.

• Engineering: Parabolic reflectors are used in satellite dishes and headlights to focus signals or light.

• Architecture: Parabolic arches are structurally strong and aesthetically pleasing, used in bridges and buildings.

• Computer Graphics: Parabolas are used to create curves and shapes in computer-aided design (CAD) software and video games.

• Economics: Quadratic functions can model various economic phenomena, such as profit maximization or cost minimization.

Advanced Analysis: Calculus Perspective

For a more in-depth understanding, we can apply calculus to analyze the graph of y = 2x² + 1:

• Derivative: The derivative of y = 2x² + 1 is dy/dx = 4x. This represents the slope of the tangent line at any point on the parabola. At the vertex (x = 0), the slope is 0, indicating a horizontal tangent.

• Second Derivative: The second derivative is d²y/dx² = 4. This constant positive value confirms the upward concavity of the parabola.

Conclusion

The graph of y = 2x² + 1, a simple yet illustrative example of a parabola, provides a rich learning opportunity. By analyzing its vertex, axis of symmetry, concavity, intercepts, and comparing it to the basic parabola y = x², we gain a deeper appreciation of quadratic functions and their geometric representation. The diverse applications of parabolas in various fields underscore their significance in mathematics and beyond. This comprehensive guide aimed to equip you with a thorough understanding of this fundamental curve, enabling you to confidently analyze and interpret its properties and applications. Remember to practice sketching parabolas with different coefficients to solidify your understanding.