Graph Y 2x 2 1

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

This article provides a comprehensive exploration of the quadratic function y = 2x² + 2x + 1, covering its graphical representation, key features, and analytical methods for understanding its behavior. We'll delve into the process of graphing this function, examining its vertex, axis of symmetry, intercepts, and overall shape. Understanding this function will solidify your grasp of quadratic equations and their graphical interpretations.

Introduction to Quadratic Functions and Their Graphs

A quadratic function is a polynomial function of degree two, generally represented in the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is always a parabola, a U-shaped curve that opens upwards if a > 0 (like our example) and downwards if a < 0. Understanding the properties of a parabola is crucial for analyzing and interpreting quadratic functions. The equation y = 2x² + 2x + 1 falls squarely within this category, with a = 2, b = 2, and c = 1. This means the parabola will open upwards.

1. Finding the Vertex of the Parabola

The vertex represents the lowest (or highest, if the parabola opens downwards) point on the parabola. For a quadratic function in the form y = ax² + bx + c, the x-coordinate of the vertex is given by the formula: x = -b / 2a. In our case, a = 2 and b = 2, so the x-coordinate of the vertex is:

x = -2 / (2 * 2) = -2 / 4 = -0.5

To find the y-coordinate, we substitute this x-value back into the original equation:

y = 2(-0.5)² + 2(-0.5) + 1 = 2(0.25) - 1 + 1 = 0.5

Therefore, the vertex of the parabola is (-0.5, 0.5). This point is crucial because it represents the minimum value of the function.

2. Determining the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is simply x = the x-coordinate of the vertex. In our case, the axis of symmetry is x = -0.5. This line serves as a useful reference point when sketching the graph.

3. Calculating the y-intercept

The y-intercept is the point where the graph intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the equation gives:

y = 2(0)² + 2(0) + 1 = 1

Therefore, the y-intercept is (0, 1).

4. Finding the x-intercepts (Roots or Zeros)

The x-intercepts, also known as roots or zeros, are the points where the graph intersects the x-axis (where y = 0). To find them, we set y = 0 and solve the quadratic equation:

2x² + 2x + 1 = 0

We can use the quadratic formula to solve for x:

x = [-b ± √(b² - 4ac)] / 2a

Substituting our values (a = 2, b = 2, c = 1):

x = [-2 ± √(2² - 4 * 2 * 1)] / (2 * 2) = [-2 ± √(-4)] / 4

Notice that the discriminant (b² - 4ac = -4) is negative. This means there are no real x-intercepts. The parabola does not intersect the x-axis. This is consistent with the fact that the vertex lies above the x-axis and the parabola opens upwards. The roots are complex numbers.

5. Sketching the Graph

Now that we have the vertex (-0.5, 0.5), the axis of symmetry (x = -0.5), and the y-intercept (0, 1), we can sketch the graph. Because the parabola opens upwards (a > 0) and has no x-intercepts, the graph will be a U-shaped curve entirely above the x-axis. The vertex represents the minimum point of the curve. You can plot additional points by substituting various x-values into the equation to get corresponding y-values for a more accurate sketch.

6. Understanding the Discriminant and Nature of Roots

The discriminant (b² - 4ac) plays a crucial role in determining the nature of the roots (x-intercepts) of a quadratic equation.

• If b² - 4ac > 0: The equation has two distinct real roots. The parabola intersects the x-axis at two points.
• If b² - 4ac = 0: The equation has one real root (a repeated root). The parabola touches the x-axis at its vertex.
• If b² - 4ac < 0: The equation has no real roots (two complex roots). The parabola does not intersect the x-axis. This is the case for our function y = 2x² + 2x + 1.

7. Completing the Square

Another method to analyze the quadratic function is by completing the square. This method helps reveal the vertex form of the quadratic equation, which is y = a(x - h)² + k, where (h, k) is the vertex.

Let's complete the square for y = 2x² + 2x + 1:

1. Factor out the coefficient of x²: y = 2(x² + x) + 1
2. Complete the square inside the parenthesis: To complete the square for x² + x, we take half of the coefficient of x (which is 1/2) and square it ((1/2)² = 1/4). We add and subtract this value inside the parenthesis: y = 2(x² + x + 1/4 - 1/4) + 1
3. Rewrite as a perfect square: y = 2((x + 1/2)² - 1/4) + 1
4. Distribute and simplify: y = 2(x + 1/2)² - 1/2 + 1 = 2(x + 1/2)² + 1/2

This is the vertex form of the equation. The vertex is (-1/2, 1/2), which confirms our earlier calculation.

8. Using Calculus to Find the Minimum Point

For those familiar with calculus, we can use the derivative to find the x-coordinate of the vertex (minimum point). The derivative of y = 2x² + 2x + 1 is:

dy/dx = 4x + 2

Setting the derivative to zero to find critical points:

4x + 2 = 0 => x = -0.5

The second derivative is d²y/dx² = 4, which is positive, confirming that this point is a minimum. Substituting x = -0.5 back into the original equation gives the y-coordinate of the vertex, which is 0.5.

Frequently Asked Questions (FAQ)

Q: What is the range of the function y = 2x² + 2x + 1?

A: Since the parabola opens upwards and has a minimum value of 0.5 at its vertex, the range of the function is y ≥ 0.5.

Q: How can I find more points to plot on the graph?

A: You can choose various x-values and substitute them into the equation y = 2x² + 2x + 1 to find their corresponding y-values. Plotting these points will help you to accurately sketch the parabola. For example, if x = 1, y = 5; if x = -1, y = 1; if x = 2, y = 13.

Q: What is the significance of the coefficient 'a' in the quadratic equation?

A: The coefficient 'a' determines whether the parabola opens upwards (a > 0) or downwards (a < 0) and also affects the steepness of the parabola. A larger absolute value of 'a' indicates a narrower parabola, while a smaller absolute value indicates a wider parabola.

Q: Are there any applications of quadratic functions in real life?

A: Quadratic functions have numerous applications in various fields, including physics (projectile motion), engineering (designing parabolic arches), and economics (modeling cost functions).

Conclusion

This detailed analysis of the graph of y = 2x² + 2x + 1 demonstrates the various methods available for understanding and representing quadratic functions. By understanding the vertex, axis of symmetry, intercepts, and discriminant, we can accurately sketch the graph and analyze its behavior. The application of both algebraic and calculus techniques provides a comprehensive approach to problem-solving and reinforces core mathematical concepts. Remember that practicing these methods with different quadratic equations will further solidify your understanding. The key is to systematically apply the techniques outlined above to any given quadratic equation, allowing you to confidently analyze and visualize its graphical representation.