Rewriting $y=\frac{2}{3}(x-6)^2-3$ In Standard Form A Step-by-Step Guide

$$

Have you ever encountered a quadratic equation in vertex form and wondered how to transform it into the standard form? Well, you're not alone! Many students find this process a bit tricky, but don't worry, guys, we're here to break it down step by step. In this article, we'll focus on rewriting the quadratic equation $y=\frac{2}{3}(x-6)^2-3$ from vertex form into standard form. We'll explore the underlying concepts, walk through the algebraic manipulations, and provide clear explanations to help you master this skill. So, let's dive in and unravel the mystery of quadratic transformations!

Understanding Quadratic Equations: Vertex Form vs. Standard Form

Before we jump into the transformation, let's quickly review the two forms of quadratic equations we'll be working with:

• Vertex Form: $y = a(x - h)^2 + k$
  • Here, $(h, k)$ represents the vertex of the parabola, and 'a' determines the direction and stretch of the parabola.
• Standard Form: $y = ax^2 + bx + c$
  • In this form, 'a' still determines the direction and stretch, while 'b' and 'c' influence the position and y-intercept of the parabola.

Our goal is to take the equation from vertex form, which readily shows the vertex, and convert it into standard form, which is often more convenient for certain algebraic manipulations and finding the y-intercept. This transformation involves expanding the squared term and simplifying the expression.

Why Convert Between Forms?

You might be wondering, “Why bother converting between these forms?” Well, each form offers unique insights and advantages:

• Vertex Form:
  • Directly reveals the vertex (h, k): This is super useful for quickly identifying the maximum or minimum point of the parabola.
  • Makes graphing easier: Knowing the vertex makes it simpler to sketch the parabola.
• Standard Form:
  • Easy to find the y-intercept: The constant term 'c' directly gives the y-intercept (0, c).
  • Convenient for using the quadratic formula: When solving for the roots (x-intercepts), the standard form is essential.
  • Facilitates algebraic manipulations: Standard form is often preferred for completing the square or other algebraic operations.

So, understanding how to convert between these forms gives you a powerful toolkit for analyzing and working with quadratic equations. It's like being bilingual in the language of parabolas! Now that we understand the importance of this transformation, let's get into the nitty-gritty of how to do it.

Step-by-Step Guide to Rewriting $y=\frac{2}{3}(x-6)^2-3$ in Standard Form

Okay, guys, let's tackle the main problem! We have the equation $y=\frac{2}{3}(x-6)^2-3$ in vertex form, and we want to rewrite it in the standard form ($y = ax^2 + bx + c$). Here’s how we'll do it:

Step 1: Expand the Squared Term

The first thing we need to do is expand the $(x-6)^2$ term. Remember, squaring a binomial means multiplying it by itself: $(x-6)^2 = (x-6)(x-6)$. We can use the FOIL method (First, Outer, Inner, Last) to expand this:

• First: $x * x = x^2$
• Outer: $x * -6 = -6x$
• Inner: $-6 * x = -6x$
• Last: $-6 * -6 = 36$

Combining these, we get: $(x-6)^2 = x^2 - 6x - 6x + 36 = x^2 - 12x + 36$

Step 2: Distribute the $\frac{2}{3}$

Now we need to multiply the expanded term by the $\frac{2}{3}$ outside the parentheses:

$\frac{2}{3}(x^2 - 12x + 36) = \frac{2}{3}x^2 - \frac{2}{3}(12x) + \frac{2}{3}(36)$

Let's simplify each term:

• $\frac{2}{3}x^2$ remains as is.
• $\frac{2}{3}(12x) = \frac{24}{3}x = 8x$
• $\frac{2}{3}(36) = \frac{72}{3} = 24$

So, we now have: $\frac{2}{3}x^2 - 8x + 24$

Step 3: Combine Constant Terms

Don't forget about the $-3$ at the end of the original equation! We need to bring that into the mix:

$y = \frac{2}{3}x^2 - 8x + 24 - 3$

Now, combine the constant terms:

$24 - 3 = 21$

Step 4: Write the Equation in Standard Form

Finally, we can write the equation in standard form by putting all the pieces together:

$y = \frac{2}{3}x^2 - 8x + 21$

And there you have it! We've successfully rewritten the equation from vertex form to standard form. Our final answer is $y = \frac{2}{3}x^2 - 8x + 21$.

Let's Check Our Work

To make sure we didn't make any mistakes, let's briefly check our work. We can compare the y-intercepts of both forms. In the standard form, the y-intercept is the constant term, which is 21.

In the vertex form, we can find the y-intercept by setting $x = 0$:

$y = \frac{2}{3}(0-6)^2 - 3 = \frac{2}{3}(36) - 3 = 24 - 3 = 21$

The y-intercepts match, which gives us confidence in our answer!

Analyzing the Options: Which One is Correct?

Now that we've done the work ourselves, let's look at the options provided and see which one matches our result:

A. $y=\frac{2}{3} x^2-8 x+21$ B. $y=\frac{2}{3} x^2-8 x+24$ C. $y=\frac{2}{3} x^2-12 x+36$ D. $y=\frac{2}{8} x^2-12 x+33$

Comparing our result, $y = \frac{2}{3}x^2 - 8x + 21$, with the options, we can clearly see that option A is the correct answer. The other options have different coefficients for the $x$ term or the constant term.

Common Mistakes to Avoid When Rewriting Quadratic Equations

Guys, when rewriting quadratic equations, it's easy to stumble if you're not careful. Here are some common pitfalls to watch out for:

1. Forgetting to Distribute: Make sure you distribute the 'a' value (in our case, $\frac{2}{3}$) to all terms inside the parentheses after expanding the squared term. It’s a classic mistake to only multiply it with the $x^2$ term.
2. Incorrectly Expanding the Squared Term: Remember that $(x - h)^2$ is not the same as $x^2 - h^2$. You need to use FOIL or the binomial expansion to correctly expand $(x - h)(x - h)$.
3. Sign Errors: Pay close attention to signs, especially when expanding and combining terms. A small sign error can throw off the entire calculation.
4. Combining Unlike Terms: Only combine terms that have the same variable and exponent. For example, you can combine $-8x$ with another 'x' term, but not with a constant or an $x^2$ term.
5. Skipping Steps: While it might be tempting to rush through the process, skipping steps increases the likelihood of making a mistake. Take your time and write out each step clearly.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence in rewriting quadratic equations.

Practice Makes Perfect: More Examples to Try

Alright, you've seen how to rewrite $y=\frac{2}{3}(x-6)^2-3$ in standard form. But to truly master this skill, you need practice! Here are a few more examples you can try on your own:

1. $y = 2(x + 3)^2 - 1$
2. $y = -\frac{1}{2}(x - 4)^2 + 5$
3. $y = 3(x - 1)^2 + 2$

For each of these, follow the same steps we used earlier:

1. Expand the squared term.
2. Distribute the coefficient.
3. Combine constant terms.
4. Write the equation in standard form.

After you've worked through these examples, you'll be well on your way to becoming a quadratic equation pro! Remember, the key is to practice consistently and break down the problem into manageable steps. And if you get stuck, don't hesitate to review the steps we covered in this article.

Conclusion: Mastering Quadratic Transformations

Guys, rewriting quadratic equations from vertex form to standard form is a fundamental skill in algebra. It allows you to analyze and manipulate quadratic functions more effectively. By understanding the steps involved—expanding the squared term, distributing, and combining like terms—you can confidently transform equations and solve a variety of problems.

In this article, we walked through a detailed example, $y=\frac{2}{3}(x-6)^2-3$, and showed you exactly how to convert it to standard form. We also discussed common mistakes to avoid and provided additional examples for practice. With consistent effort and a clear understanding of the process, you can master this skill and excel in your algebra studies.

So, keep practicing, stay curious, and remember that every mathematical challenge is an opportunity to learn and grow. You've got this!