write in vertex form y 8 x 2

Guri Bee

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19-10-2024

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Unlocking the Secrets of Quadratic Functions: Writing in Vertex Form (y = 8x^2)

The world of mathematics is full of fascinating patterns, and one of the most captivating is the quadratic function. These functions, characterized by their parabolic shape, are found everywhere from projectile motion to optimizing business strategies. A crucial tool for understanding and manipulating quadratic functions is the vertex form.

What is Vertex Form?

The vertex form of a quadratic equation is written as:

y = a(x - h)^2 + k

Where:

• (h, k) represents the coordinates of the vertex (the highest or lowest point on the parabola)
• a determines the direction and steepness of the parabola:
  • If a is positive, the parabola opens upwards.
  • If a is negative, the parabola opens downwards.
  • The larger the absolute value of a, the steeper the parabola.

Transforming to Vertex Form

Let's explore how to rewrite a quadratic equation in vertex form, using the example of y = 8x^2.

1. Identifying the 'a' Value

In our equation, the coefficient of the x² term is 8. Therefore, a = 8.

2. Finding the Vertex

The vertex of the parabola lies on the line of symmetry, which is defined by the equation x = -b/2a. In our case, since there is no x term, b = 0. Hence, the line of symmetry is x = 0.

To find the y-coordinate of the vertex, substitute x = 0 into the original equation:

y = 8(0)^2 = 0

Therefore, the vertex of the parabola is (0, 0).

3. Completing the Square

Since we already know the vertex, we can directly write the equation in vertex form:

y = 8(x - 0)^2 + 0

4. Simplifying the Equation

Simplifying the equation, we get:

y = 8x^2

The Power of Vertex Form

Writing a quadratic function in vertex form provides several benefits:

• Easy Identification of the Vertex: The vertex form immediately reveals the coordinates of the vertex, facilitating analysis and graphing.
• Understanding Transformations: By comparing the vertex form to the standard form, we can identify shifts, stretches, and reflections of the parabola.
• Solving for Extrema: The vertex form helps determine the maximum or minimum value of the quadratic function.

Example:

Let's say you're a business owner trying to maximize profit. Your profit function is modeled by the equation P(x) = -2x^2 + 12x - 10, where x represents the number of units sold.

To find the maximum profit, you can rewrite the equation in vertex form:

P(x) = -2(x - 3)^2 + 8

From the vertex form, we see that the maximum profit of 8 is achieved when 3 units are sold.

Conclusion

Understanding vertex form is a crucial step in mastering quadratic functions. It unlocks deeper insights into their behavior and provides a powerful tool for solving various problems in mathematics and real-world applications. So, the next time you encounter a quadratic equation, try rewriting it in vertex form—it might reveal a hidden treasure of knowledge!