Solving Quadratic Equations Finding Equivalent Forms

Which equation is equivalent to the formula below?

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

A. $k=y+(x-h)^2$ B. $a=\frac{y-k}{(x-h)^2}$ C. $x= \pm \sqrt{\frac{y-k}{a}}+h$ D. $h=x-\left(\frac{y-k}{a}\right)^2$

In the realm of mathematics, quadratic equations play a pivotal role. They're not just abstract concepts confined to textbooks; they have tangible applications in various fields, from physics to engineering. Understanding quadratic equations and their transformations is crucial for solving a wide array of problems. This article delves into the intricacies of manipulating quadratic equations, using a specific example to illustrate the process. We will dissect the given equation, explore the various transformations, and ultimately identify the correct equivalent equation. This comprehensive exploration aims to enhance your understanding of quadratic equations and equip you with the skills to tackle similar problems with confidence.

Deconstructing the Quadratic Equation

The given equation, $y = a(x - h)^2 + k$, represents the vertex form of a quadratic equation. This form is particularly useful because it readily reveals the vertex of the parabola, which is the point $(h, k)$. The parameter 'a' determines the direction and the 'steepness' of the parabola. A positive 'a' indicates that the parabola opens upwards, while a negative 'a' indicates that it opens downwards. The absolute value of 'a' dictates how stretched or compressed the parabola is; a larger absolute value results in a narrower parabola, and a smaller absolute value results in a wider parabola. The values h and k are translations or shifts. The h value is a horizontal shift, and the k value is a vertical shift. Understanding each component of this equation is paramount to manipulating it effectively. When solving for a variable, it's essential to follow the order of operations in reverse, isolating the term containing the desired variable step by step. We can achieve the equivalent equations, by algebraic manipulation, adding, subtracting, multiplying, or dividing both sides of the equation by the same value to solve for one variable in terms of the others. It involves isolating the desired variable on one side of the equation while keeping the equation balanced. This process often requires careful application of algebraic principles, such as the distributive property and inverse operations.

Step-by-Step Transformation

To determine the equivalent equation, we need to manipulate the original equation, $y = a(x - h)^2 + k$, and isolate each of the variables listed in the answer choices, one at a time. Let's examine each option systematically.

Option A: Isolating 'k'

To isolate 'k', we need to remove the term $a(x - h)^2$ from the right side of the equation. We can achieve this by subtracting $a(x - h)^2$ from both sides:

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

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

This can be rewritten as:

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

Comparing this with option A, $k = y + (x - h)^2$, we can see that they are not equivalent because there is a minus sign missing and also the variable 'a'. Therefore, option A is incorrect.

Option B: Isolating 'a'

To isolate 'a', we first need to isolate the term containing 'a', which is $a(x - h)^2$. We can do this by subtracting 'k' from both sides:

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

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

Next, we divide both sides by $(x - h)^2$:

$\frac{y - k}{(x - h)^2} = \frac{a(x - h)^2}{(x - h)^2}$

$\frac{y - k}{(x - h)^2} = a$

This can be rewritten as:

$a = \frac{y - k}{(x - h)^2}$

Comparing this with option B, $a = \frac{y - k}{(x - h)^2}$, we can see that they are equivalent. Therefore, option B is a potential correct answer.

Option C: Isolating 'x'

To isolate 'x', we again start by subtracting 'k' from both sides:

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

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

Next, we divide both sides by 'a':

$\frac{y - k}{a} = \frac{a(x - h)^2}{a}$

$\frac{y - k}{a} = (x - h)^2$

Now, we take the square root of both sides:

$\pm \sqrt{\frac{y - k}{a}} = \sqrt{(x - h)^2}$

$\pm \sqrt{\frac{y - k}{a}} = x - h$

Finally, we add 'h' to both sides:

$x = \pm \sqrt{\frac{y - k}{a}} + h$

Comparing this with option C, $x = \pm \sqrt{\frac{y - k}{a}} - h$, we can see that the sign of 'h' is different. Therefore, option C is incorrect.

Option D: Isolating 'h'

From our work in isolating 'x' in option C, we have the equation:

$\pm \sqrt{\frac{y - k}{a}} = x - h$

To isolate 'h', we can subtract 'x' from both sides:

$\pm \sqrt{\frac{y - k}{a}} - x = -h$

Now, multiply both sides by -1:

$x \mp \sqrt{\frac{y - k}{a}} = h$

This can be rewritten as:

$h = x \mp \sqrt{\frac{y - k}{a}}$

Comparing this with option D, $h = x - \left(\frac{y - k}{a}\right)^2$, we can see that they are not equivalent. Therefore, option D is incorrect.

Conclusion

After carefully manipulating the original equation and comparing the results with the given options, we have determined that only option B, $a = \frac{y - k}{(x - h)^2}$, is equivalent to the given formula. This exercise highlights the importance of understanding algebraic manipulations and the properties of quadratic equations. By mastering these skills, you can confidently solve a wide range of mathematical problems and gain a deeper appreciation for the elegance and power of mathematics. Remember, practice is key to honing your skills in manipulating equations. Work through various examples, and don't hesitate to seek clarification when needed. The journey of learning mathematics is a rewarding one, filled with exciting discoveries and the satisfaction of solving complex problems. Continue to explore, question, and practice, and you will undoubtedly excel in your mathematical endeavors.