Quadratic Function Analysis Unveiling Y=2x²-5x+1

Hey guys! Today, we're diving deep into the fascinating world of quadratic functions, specifically the function y=2x²-5x+1. We'll dissect this equation, exploring its various components and how they influence the graph, or parabola, it represents. Get ready to sharpen your pencils and unleash your inner mathematician as we embark on this journey together!

The Anatomy of a Quadratic Function

Before we jump into the specifics of our function, let's first understand the general form of a quadratic equation. A quadratic function is typically expressed as y = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. Each of these coefficients plays a crucial role in shaping the parabola. The 'a' coefficient dictates the parabola's concavity – whether it opens upwards (if a is positive) or downwards (if a is negative). In our case, a=2, which is positive, so we know our parabola will be smiling upwards. The magnitude of 'a' also affects the width of the parabola; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider one. So, in this case, the a value of 2 makes the parabola vertically stretched compared to the basic quadratic function y = x².

The 'b' coefficient is a bit trickier; it influences the position of the parabola's axis of symmetry and its horizontal shift. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. The x-coordinate of the axis of symmetry is given by the formula x = -b / 2a. In our function, b=-5, so this will play a significant role in determining the parabola's position on the coordinate plane. The interplay between 'a' and 'b' dictates whether the parabola shifts left or right. Finally, the 'c' coefficient represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. In our function, c=1, meaning the parabola will cross the y-axis at the point (0, 1). This is a crucial point for sketching the graph and understanding the function's behavior. To truly grasp the function, we need to delve deeper into these elements, exploring their combined effect on the parabola's shape and position.

Delving into y=2x²-5x+1: A Detailed Examination

Now, let's apply our knowledge to the specific function y=2x²-5x+1. We've already established that a=2, b=-5, and c=1. This is where the fun really begins! Using these values, we can determine key features of the parabola. First, let's find the axis of symmetry. Using the formula x = -b / 2a, we get x = -(-5) / (22) = 5/4 = 1.25*. This means the vertical line x=1.25 is the parabola's axis of symmetry, dividing it into two mirror images. This value, 1.25, is crucial as it also represents the x-coordinate of the vertex, the parabola's turning point. To find the y-coordinate of the vertex, we substitute x=1.25 back into the original equation: y = 2(1.25)² - 5(1.25) + 1 = 2(1.5625) - 6.25 + 1 = 3.125 - 6.25 + 1 = -2.125. Therefore, the vertex of the parabola is at the point (1.25, -2.125). This is the minimum point of the parabola since the coefficient a is positive, signifying that the parabola opens upwards.

Knowing the vertex is vital, but we can extract further insights by calculating the discriminant. The discriminant, given by the formula Δ = b² - 4ac, provides information about the nature and number of real roots (x-intercepts) of the quadratic equation. For our function, the discriminant is Δ = (-5)² - 4(2)(1) = 25 - 8 = 17. Since the discriminant is positive, we know that the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points. The values of these roots can be calculated using the quadratic formula: x = (-b ± √Δ) / 2a. Substituting our values, we get x = (5 ± √17) / 4. Calculating these values gives us approximately x₁ ≈ 0.28 and x₂ ≈ 2.28. These are the x-intercepts of our parabola, providing us with two more crucial points for sketching the graph and understanding the function's behavior. By determining the vertex, the axis of symmetry, and the x-intercepts, we have established a robust foundation for visualizing and analyzing the quadratic function y=2x²-5x+1.

Graphing the Parabola: Visualizing the Function

Now that we've calculated the key features of the quadratic function, let's visualize it by sketching its graph. Remember, the graph of a quadratic function is a parabola, a U-shaped curve. We already know the vertex is at (1.25, -2.125), which is the lowest point on the curve. We also know the axis of symmetry is the vertical line x = 1.25, which helps us understand the symmetry of the parabola around this line. Furthermore, we've calculated the x-intercepts to be approximately 0.28 and 2.28, meaning the parabola crosses the x-axis at these two points. And, of course, we know the y-intercept is (0, 1), where the parabola intersects the y-axis. Plotting these points – the vertex, the x-intercepts, and the y-intercept – on a coordinate plane gives us a good starting point for sketching the parabola. Since we know the parabola opens upwards (a is positive), we can sketch a smooth, U-shaped curve that passes through these points and is symmetrical about the axis of symmetry.

To make our sketch even more accurate, we can find a few additional points on the parabola. For example, we can choose an x-value, say x=3, and substitute it into the equation to find the corresponding y-value: y = 2(3)² - 5(3) + 1 = 18 - 15 + 1 = 4. So, the point (3, 4) lies on the parabola. Because of the symmetry, we know that the point symmetrical to (3, 4) across the axis of symmetry x=1.25 will also be on the parabola. The x-coordinate of this symmetrical point will be 1.25 - (3 - 1.25) = -0.5, and the y-coordinate will be the same, 4. Thus, the point (-0.5, 4) also lies on the parabola. By plotting these extra points, we can refine our sketch and obtain a more accurate representation of the quadratic function. The graph visually reinforces all our calculations, giving us a comprehensive understanding of the function's behavior. The vertex represents the minimum value of the function, and the x-intercepts indicate the values of x for which the function equals zero. The y-intercept provides a clear indication of the function's value at x=0. The parabola's symmetry allows us to make predictions about the function's behavior for values of x equidistant from the axis of symmetry. Ultimately, graphing the parabola bridges the gap between the algebraic representation of the function and its visual manifestation, enriching our overall understanding.

Applications of Quadratic Functions: Beyond the Classroom

Okay, guys, we've thoroughly analyzed the quadratic function y=2x²-5x+1, but you might be wondering,