Rewriting $y=2x^2-8x+9$ In Vertex Form A Step-by-Step Guide

In the realm of quadratic equations, the vertex form holds a special significance. It provides a clear view of the parabola's vertex, which is crucial for understanding the graph's behavior. Today, we'll dive into the process of transforming the given quadratic equation, $y = 2x^2 - 8x + 9$, into its vertex form. This transformation not only reveals the vertex but also offers insights into the parabola's symmetry and direction.

Understanding Vertex Form

Before we embark on the transformation journey, let's first grasp the essence of vertex form. A quadratic equation in vertex form is expressed as:

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

where:

• $(h, k)$ represents the vertex of the parabola.
• $a$ determines the direction and width of the parabola. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards. The magnitude of $a$ dictates the parabola's width; a larger magnitude corresponds to a narrower parabola.

With this understanding, our goal is to manipulate the given equation, $y = 2x^2 - 8x + 9$, to match this vertex form structure. This involves a technique known as completing the square.

Completing the Square: The Key to Vertex Form

Completing the square is a powerful algebraic technique that allows us to rewrite a quadratic expression in a form that reveals a squared term. This is precisely what we need to achieve vertex form. Let's break down the steps:

1. Factor out the leading coefficient:

   Our equation is $y = 2x^2 - 8x + 9$. We begin by factoring out the coefficient of the $x^2$ term, which is 2, from the first two terms:

   $y = 2(x^2 - 4x) + 9$

   This step isolates the quadratic and linear terms within the parentheses, preparing them for the square completion process.

2. Complete the square inside the parentheses:

   Now, we focus on the expression inside the parentheses, $x^2 - 4x$. To complete the square, we need to add and subtract a constant term that will create a perfect square trinomial. This constant is determined by taking half of the coefficient of the $x$ term (which is -4), squaring it, and adding and subtracting the result. Half of -4 is -2, and squaring it gives us 4. So, we add and subtract 4 inside the parentheses:

   $y = 2(x^2 - 4x + 4 - 4) + 9$

   Notice that we've added and subtracted the same value, so we haven't changed the equation's overall value. However, we've strategically introduced a perfect square trinomial.

3. Rewrite as a squared term:

   The expression $x^2 - 4x + 4$ is a perfect square trinomial, which can be rewritten as $(x - 2)^2$. Our equation now looks like this:

   $y = 2((x - 2)^2 - 4) + 9$

   We've successfully transformed the quadratic expression into a form that includes a squared term, bringing us closer to vertex form.

4. Distribute and simplify:

   Next, we distribute the 2 back into the parentheses:

   $y = 2(x - 2)^2 - 8 + 9$

   Finally, we simplify the constant terms:

   $y = 2(x - 2)^2 + 1$

   And there we have it! The equation is now in vertex form.

Identifying the Vertex and Interpreting the Equation

By comparing our transformed equation, $y = 2(x - 2)^2 + 1$, with the general vertex form $y = a(x - h)^2 + k$, we can easily identify the vertex and other key characteristics:

• The vertex is $(h, k) = (2, 1)$. This tells us the parabola's minimum point is located at the coordinates (2, 1).
• The value of $a$ is 2. Since $a > 0$, the parabola opens upwards. The fact that $a$ is 2 indicates that the parabola is narrower than the standard parabola $y = x^2$.

The Answer

Therefore, the equation $y = 2x^2 - 8x + 9$ rewritten in vertex form is:

C. $y = 2(x - 2)^2 + 1$

Additional Insights into Vertex Form

Vertex form is not just a mathematical curiosity; it's a powerful tool for understanding and analyzing quadratic functions. Here are some additional benefits of having an equation in vertex form:

• Finding the axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. In vertex form, the axis of symmetry is simply the vertical line $x = h$. In our example, the axis of symmetry is $x = 2$.
• Determining the range: The range of a quadratic function is the set of all possible $y$-values. If the parabola opens upwards (as in our example), the range is all $y$-values greater than or equal to the $y$-coordinate of the vertex. If the parabola opens downwards, the range is all $y$-values less than or equal to the $y$-coordinate of the vertex. In our case, the range is $y ≥ 1$.
• Graphing the parabola: Vertex form makes it easy to graph a parabola. You can start by plotting the vertex, then use the axis of symmetry to find a corresponding point on the other side of the vertex. The value of $a$ helps you determine the parabola's width and direction.

Conclusion

Mastering the technique of completing the square and transforming quadratic equations into vertex form is a valuable skill in algebra. It provides a deeper understanding of parabolas and their properties, enabling us to analyze and interpret quadratic functions more effectively. By understanding vertex form, we gain insights into the vertex, axis of symmetry, range, and overall shape of the parabola. This knowledge empowers us to solve a wide range of problems involving quadratic equations and their graphical representations.

Remember, the key to success in mathematics is practice. So, try transforming different quadratic equations into vertex form, and you'll become more comfortable with the process and its applications.