Graph Y 3x 2 1

Unveiling the Secrets of the Graph y = 3x² + 2x + 1: A Comprehensive Guide

This article delves deep into the fascinating world of the quadratic function represented by the equation y = 3x² + 2x + 1. We'll explore its various properties, graphing techniques, and real-world applications, providing a comprehensive understanding suitable for students and enthusiasts alike. This exploration will cover key aspects such as finding the vertex, axis of symmetry, intercepts, and discriminant, ultimately enabling you to fully visualize and interpret this quadratic function.

Meta Description: Dive into a comprehensive analysis of the quadratic function y = 3x² + 2x + 1. Learn how to graph it, understand its properties like vertex, axis of symmetry, intercepts, and discriminant, and explore its real-world applications.

Understanding the Quadratic Equation

The equation y = 3x² + 2x + 1 is a quadratic equation, meaning it's a polynomial of degree two. The general form of a quadratic equation is ax² + bx + c = y, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our case, a = 3, b = 2, and c = 1. The fact that 'a' (the coefficient of x²) is positive indicates that the parabola opens upwards, meaning it has a minimum point (vertex).

Finding the Vertex: The Turning Point

The vertex of a parabola represents its minimum or maximum point. For a quadratic equation in the form ax² + bx + c, the x-coordinate of the vertex is given by: x = -b / 2a. Let's apply this to our equation:

x = -2 / (2 * 3) = -1/3

Now, substitute this x-value back into the original equation to find the y-coordinate:

y = 3(-1/3)² + 2(-1/3) + 1 = 1/3 - 2/3 + 1 = 2/3

Therefore, the vertex of the parabola is located at (-1/3, 2/3). This point is crucial for accurately graphing the function.

Determining the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images. Its equation is simply x = the x-coordinate of the vertex. In our case, the axis of symmetry is x = -1/3. Understanding the axis of symmetry helps in plotting points symmetrically, making graphing more efficient.

Calculating the 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:

y = 3(0)² + 2(0) + 1 = 1

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

Finding the x-intercepts (Roots)

The x-intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis (where y = 0). To find these, we need to solve the quadratic equation 3x² + 2x + 1 = 0. We can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Substituting our values:

x = [-2 ± √(2² - 4 * 3 * 1)] / (2 * 3) = [-2 ± √(-8)] / 6

Notice that the discriminant (b² - 4ac = -8) is negative. This means that the parabola does not intersect the x-axis; there are no real roots. The parabola lies entirely above the x-axis.

Understanding the Discriminant

The discriminant (b² - 4ac) provides valuable information about the nature of the roots of a quadratic equation:

Discriminant > 0: Two distinct real roots (two x-intercepts).
Discriminant = 0: One real root (the parabola touches the x-axis at one point).
Discriminant < 0: No real roots (the parabola does not intersect the x-axis), as in our case.

Graphing the Parabola

Now, armed with the vertex, axis of symmetry, y-intercept, and the knowledge that there are no x-intercepts, we can accurately graph the parabola y = 3x² + 2x + 1. Start by plotting the vertex (-1/3, 2/3) and the y-intercept (0, 1). Use the axis of symmetry (x = -1/3) to plot a symmetrical point to the y-intercept. You can also calculate additional points by substituting different x-values into the equation to get corresponding y-values. Connecting these points smoothly will reveal the upward-opening parabola. Remember to label the axes and key points on your graph.

Advanced Analysis: Completing the Square

Another method to analyze the quadratic function is by completing the square. This technique allows us to rewrite the equation in vertex form, y = a(x - h)² + k, where (h, k) represents the vertex.

Let's complete the square for y = 3x² + 2x + 1:

Factor out the coefficient of x² from the x² and x terms: y = 3(x² + (2/3)x) + 1
Take half of the coefficient of x ((2/3)/2 = 1/3), square it ((1/3)² = 1/9), and add and subtract this value inside the parenthesis: y = 3(x² + (2/3)x + 1/9 - 1/9) + 1
Rewrite the expression as a perfect square: y = 3((x + 1/3)² - 1/9) + 1
Distribute the 3 and simplify: y = 3(x + 1/3)² - 1/3 + 1 = 3(x + 1/3)² + 2/3

This vertex form confirms our previously calculated vertex of (-1/3, 2/3).

Real-World Applications of Quadratic Functions

Quadratic equations are not merely abstract mathematical concepts; they have numerous real-world applications. They are used to model various phenomena, including:

Projectile Motion: The trajectory of a ball thrown or launched follows a parabolic path, which can be described by a quadratic equation. Variables like initial velocity and angle of projection determine the specific equation.
Area Optimization: Finding the maximum area of a rectangular enclosure given a fixed perimeter involves solving a quadratic equation.
Engineering and Physics: Quadratic equations appear in numerous engineering and physics problems related to forces, energy, and motion.
Economics: Quadratic functions can model cost, revenue, and profit functions in economic models.
Computer Graphics: Parabolas are used to create curved shapes and smooth transitions in computer graphics and animations.

Conclusion

The quadratic function y = 3x² + 2x + 1, though seemingly simple, reveals a rich tapestry of mathematical properties and real-world applications. By understanding its vertex, axis of symmetry, intercepts, and discriminant, we can accurately graph the parabola and gain valuable insights into its behavior. The techniques discussed—the quadratic formula, completing the square, and analyzing the discriminant—are fundamental tools for understanding and manipulating quadratic functions, essential for various fields of study and practical problem-solving. This comprehensive exploration underscores the importance of thoroughly understanding even seemingly basic mathematical concepts, as their applications extend far beyond the classroom. Further exploration could involve investigating transformations of this quadratic function, such as shifting, stretching, and reflecting the graph. The possibilities for deeper understanding are vast and rewarding.