Mastering Vertex Form How To Rewrite Y = 9x² + 9x - 1

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In the realm of quadratic equations, the vertex form holds a special significance. It provides a clear and concise representation of the parabola's key features, most notably its vertex – the point where the parabola changes direction. This article delves into the process of converting a quadratic equation from its standard form to the vertex form, using the specific example of y = 9x² + 9x - 1. We will explore the underlying principles, step-by-step calculations, and the geometric interpretation of the vertex form, equipping you with a comprehensive understanding of this essential mathematical concept.

Understanding Quadratic Equations and Their Forms

Before we embark on the conversion process, let's establish a solid foundation by understanding the different forms of quadratic equations. A quadratic equation is a polynomial equation of degree two, generally expressed in the following forms:

  1. Standard Form: The standard form of a quadratic equation is given by y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. This form is convenient for identifying the coefficients and applying the quadratic formula to find the roots of the equation.
  2. Vertex Form: The vertex form of a quadratic equation is given by y = a(x - h)² + k, where a is the same coefficient as in the standard form, and (h, k) represents the coordinates of the vertex of the parabola. This form directly reveals the vertex and the direction of the parabola's opening (upward if a > 0 and downward if a < 0).
  3. Factored Form: The factored form of a quadratic equation is given by y = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots (x-intercepts) of the equation. This form is useful for quickly identifying the roots of the equation.

Our focus in this article is on transforming the given equation, y = 9x² + 9x - 1, from its standard form to the vertex form. This transformation will allow us to readily identify the vertex of the parabola represented by the equation.

The Significance of Vertex Form

The vertex form, y = a(x - h)² + k, provides valuable insights into the behavior of a quadratic function. The vertex, (h, k), represents the point where the parabola attains its minimum (if a > 0) or maximum (if a < 0) value. The h-coordinate represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The k-coordinate represents the minimum or maximum value of the function.

In the context of real-world applications, the vertex form helps us solve optimization problems. For instance, if we are modeling the trajectory of a projectile, the vertex represents the highest point the projectile reaches. Similarly, if we are modeling the profit of a business, the vertex represents the point of maximum profit.

Converting from Standard Form to Vertex Form: Completing the Square

The key technique for converting a quadratic equation from standard form to vertex form is called completing the square. This method involves manipulating the equation algebraically to create a perfect square trinomial, which can then be factored into the square of a binomial.

Let's apply this technique to our equation, y = 9x² + 9x - 1:

  1. Factor out the coefficient of x²: In our equation, the coefficient of x² is 9. We factor this out from the first two terms:

    y = 9(x² + x) - 1

  2. Complete the square inside the parentheses: To complete the square, we need to add and subtract a constant term inside the parentheses. This constant is determined by taking half of the coefficient of the x term (which is 1 in this case), squaring it ( (1/2)² = 1/4 ), and adding and subtracting it:

    y = 9(x² + x + 1/4 - 1/4) - 1

  3. Rewrite the perfect square trinomial: The expression x² + x + 1/4 is a perfect square trinomial, which can be factored as (x + 1/2)²:

    y = 9((x + 1/2)² - 1/4) - 1

  4. Distribute and simplify: Distribute the 9 and simplify the expression:

    y = 9(x + 1/2)² - 9/4 - 1 y = 9(x + 1/2)² - 13/4

Therefore, the vertex form of the equation y = 9x² + 9x - 1 is y = 9(x + 1/2)² - 13/4.

Identifying the Vertex

Now that we have the equation in vertex form, y = 9(x + 1/2)² - 13/4, we can easily identify the vertex. Comparing this to the general vertex form, y = a(x - h)² + k, we see that:

  • a = 9
  • h = -1/2
  • k = -13/4

Thus, the vertex of the parabola is at the point (-1/2, -13/4). Since a = 9 is positive, the parabola opens upward, and the vertex represents the minimum point of the function.

Geometric Interpretation

The vertex form provides a clear geometric interpretation of the parabola. The vertex, (-1/2, -13/4), is the turning point of the parabola. The axis of symmetry is the vertical line x = -1/2, which passes through the vertex. The parabola is symmetric about this line.

The coefficient a = 9 determines the vertical stretch of the parabola. A larger value of a indicates a steeper parabola, while a smaller value indicates a wider parabola. In this case, the parabola is stretched vertically by a factor of 9 compared to the standard parabola y = x².

Practical Applications

The ability to convert quadratic equations to vertex form has numerous practical applications. Here are a few examples:

  • Optimization Problems: As mentioned earlier, the vertex form helps us solve optimization problems where we need to find the maximum or minimum value of a quadratic function. This is useful in various fields, such as engineering, economics, and physics.
  • Projectile Motion: The trajectory of a projectile (such as a ball thrown in the air) can be modeled using a quadratic equation. The vertex represents the highest point the projectile reaches, which is a crucial piece of information in many applications.
  • Curve Fitting: Quadratic equations can be used to fit curves to data points. The vertex form can help us understand the shape and characteristics of the curve.
  • Engineering Design: In engineering design, quadratic equations are used to model various phenomena, such as the shape of arches and suspension bridges. The vertex form helps engineers determine the critical points and dimensions of these structures.

Conclusion

Converting a quadratic equation from standard form to vertex form is a valuable skill in mathematics. The vertex form, y = a(x - h)² + k, provides a clear representation of the parabola's vertex, axis of symmetry, and direction of opening. By mastering the technique of completing the square, we can readily transform quadratic equations and unlock their geometric and practical insights. In this article, we meticulously converted the equation y = 9x² + 9x - 1 into vertex form, illustrating the process and highlighting the significance of this transformation. Understanding the vertex form empowers us to analyze quadratic functions more effectively and apply them to a wide range of real-world problems.