Solving Y = 2x² - 10x + 5 Formula And Graph Explained

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Hey guys! Let's dive into the world of quadratic equations and dissect the equation Y = 2x² - 10x + 5. This equation represents a parabola, a U-shaped curve, and we're going to explore how to find its key features and graph it. Think of it like a treasure hunt, where the treasure is understanding the equation's secrets. We'll use formulas, step-by-step methods, and even create a graph to visualize what's going on. So, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into the specifics of our equation, let's take a moment to understand the general form of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form is expressed as:

ax² + bx + c = 0

Where 'a', 'b', and 'c' are constants, and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation). In our case, we have Y = 2x² - 10x + 5, which is slightly different because it's set equal to 'Y' instead of zero. This form helps us understand the relationship between 'x' and 'Y' and allows us to graph the equation. Let's identify the coefficients in our equation:

  • a = 2
  • b = -10
  • c = 5

These coefficients are crucial for finding the vertex, the axis of symmetry, and the roots (or x-intercepts) of the parabola. The vertex is the highest or lowest point on the parabola, depending on whether the parabola opens upwards (a > 0) or downwards (a < 0). The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. The roots are the points where the parabola intersects the x-axis (where Y = 0). Think of these elements as the essential landmarks that define the shape and position of our parabolic map. Let's explore each of these elements in detail and how they relate to our specific equation.

Finding the Vertex: The Peak or Valley of the Parabola

The vertex is a key feature of a parabola, representing its minimum or maximum point. To find the vertex, we use the following formulas:

  • x-coordinate of vertex (h) = -b / 2a
  • y-coordinate of vertex (k) = f(h) (This means we plug the x-coordinate we just found back into the original equation to find the y-coordinate).

Let's apply these formulas to our equation, Y = 2x² - 10x + 5:

  1. Find the x-coordinate (h):

    • h = -(-10) / (2 * 2) = 10 / 4 = 2.5

  2. Find the y-coordinate (k):

    • k = 2(2.5)² - 10(2.5) + 5
    • k = 2(6.25) - 25 + 5
    • k = 12.5 - 25 + 5
    • k = -7.5

Therefore, the vertex of our parabola is (2.5, -7.5). This point is the lowest point on our parabola because the coefficient 'a' (2) is positive, indicating that the parabola opens upwards. The vertex is like the keystone of an arch, the crucial point that dictates the overall structure. In our case, it tells us where the parabola bottoms out before curving back upwards.

Axis of Symmetry: The Mirror Line

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation for the axis of symmetry is simply:

x = h

Where 'h' is the x-coordinate of the vertex. In our case, we found that the x-coordinate of the vertex is 2.5, so the axis of symmetry is:

x = 2.5

Imagine folding the parabola along this line; the two halves would perfectly overlap. This symmetry is a fundamental property of parabolas and makes them visually appealing and mathematically predictable. The axis of symmetry acts like a mirror, reflecting one side of the parabola onto the other.

Finding the Roots (x-intercepts): Where the Parabola Crosses the X-Axis

The roots, also known as x-intercepts, are the points where the parabola intersects the x-axis (Y = 0). To find the roots, we need to solve the quadratic equation when Y = 0:

2x² - 10x + 5 = 0

We can use the quadratic formula to solve for 'x':

x = (-b ± √(b² - 4ac)) / 2a

Let's plug in our coefficients (a = 2, b = -10, c = 5):

  • x = (10 ± √((-10)² - 4 * 2 * 5)) / (2 * 2)
  • x = (10 ± √(100 - 40)) / 4
  • x = (10 ± √60) / 4
  • x = (10 ± 2√15) / 4
  • x = (5 ± √15) / 2

So, we have two roots:

  • x₁ = (5 + √15) / 2 ≈ 4.44
  • x₂ = (5 - √15) / 2 ≈ 0.56

These are the points where our parabola crosses the x-axis. They give us a crucial sense of the parabola's horizontal position and its relationship to the x-axis. The roots are like the parabola's anchors on the x-axis, marking where the curve intersects this important reference line.

Graphing the Quadratic Equation

Now that we have all the key information – the vertex, axis of symmetry, and roots – we can graph the quadratic equation Y = 2x² - 10x + 5. Here's how:

  1. Plot the vertex: We found the vertex to be (2.5, -7.5). Locate this point on the coordinate plane and mark it.
  2. Draw the axis of symmetry: Draw a vertical dashed line through x = 2.5. This line will help you create a symmetrical parabola.
  3. Plot the roots: We found the roots to be approximately 4.44 and 0.56. Mark these points on the x-axis.
  4. Find additional points: To get a better sense of the parabola's shape, you can choose a few additional x-values and calculate the corresponding y-values. For example:
    • If x = 0, Y = 2(0)² - 10(0) + 5 = 5. Plot the point (0, 5).
    • Since the parabola is symmetrical, there will be a corresponding point on the other side of the axis of symmetry. The point symmetrical to (0, 5) is (5, 5). Plot this point.
    • If x = 1, Y = 2(1)² - 10(1) + 5 = -3. Plot the point (1, -3).
    • The point symmetrical to (1, -3) is (4, -3). Plot this point.
  5. Draw the parabola: Sketch a smooth U-shaped curve that passes through the plotted points. The curve should be symmetrical about the axis of symmetry and should open upwards since 'a' is positive.

By following these steps, you'll create a visual representation of the quadratic equation Y = 2x² - 10x + 5. The graph vividly illustrates the relationship between 'x' and 'Y' and how the vertex, axis of symmetry, and roots define the parabola's shape and position.

Conclusion

We've successfully navigated the world of the quadratic equation Y = 2x² - 10x + 5. We've learned how to identify the coefficients, calculate the vertex and axis of symmetry, find the roots, and ultimately, graph the equation. Understanding these steps allows you to analyze any quadratic equation and visualize its corresponding parabola. Remember, guys, practice makes perfect! The more you work with quadratic equations, the more comfortable you'll become with them. So, keep exploring, keep learning, and keep graphing!