Rewriting an equation in another form, or rewriting an equation, can be useful for solving for a variable. Vertex form is one way to rewrite equations that can be useful in solving for a variable.
An equation in vertex form has the variable listed as a single value, called the vertex, and values above and below the variable that are added or subtracted from it to make it match the sides of the triangle formed by the sides of the equation.
Solving an equation in vertex form requires finding what are called incantations. An incantation is simply a set of steps to take to solve an equation in vertex form. These incantations can be remembered easily if you think of the term “incantation” as referring to how you chant these steps to solve the problem.
This article will discuss how to rewrite y = –6×2 + 3x + 2 in vertex form and how to solve such an equation in vertex form.
Identify the y-intercept
The y-intercept of an equation is the value when x equals zero. For example, if y = 2x + 1 then 1 is the y-intercept because when x equals zero, then y equals one.
Bullet point: Rewrite in vertex form
To rewrite the equation in vertex form, first solve for the variable x and then put that value on the x-axis. To solve for x, take the opposite of what is in front of it and divide it by what is behind it.
In this case, -6×2 + 3x + 2 is rewritten as (x – 2)2. The square on both sides of the variable indicates that this equation is in vertex form.
Rewrite the equation in y = a(x – h) + k format
Once you have the vertex form, you can rewrite the equation in y = a(x – h) + k format. This step is necessary because you will need to know how to write the graph of the function in vertex form in order to solve for k.
To do this, first write the equation in y = a(x – h) format. Then, take the a(x – h) and multiply it by -1, then add h. Finally, add k to the end of the variable. This will solve for k, and you can ignore it since it is not used in the graph of the function.
Example: To rewrite Y = -6×2 + 3x + 2 in vertex form, first write y = a(x – h) + k. Then, take out a(x – h) and put it over -1, then add 2h + k.
Find where the line crosses the y-axis
To rewrite the equation in vertex form, start by putting the y-intercept in for y. Since the y-intercept is 2, put 2 for y.
Then solve for x by putting –6×2 + 3x + 2 into an equation solver like one found in Google Docs. You should get –6x = 0 as the solution for x, so put 0 for x in the y = 2 equation.
Check to make sure that the variables are on opposite sides of the equation by looking at the coefficients. The coefficient of y is –6, so make sure that –6 is not a variable in the new equation. The coefficient of x is 0, so make sure that 0 is not a variable in the new equation.
The only variable left is y, so put an equals sign before it and replace it with −6×0 + 3×0 + 2 = −6(0) + 3(0) + 2=2 which is true.
Identify a, b, and c values
In order to rewrite the equation in vertex form, you must first identify the a, b, and c values. The a value is the y-coordinate of the vertex, the b value is the x-coordinate of the vertex, and the c value is the magnitude of the vertex (the coefficient of x).
The a value can be any non-zero number. The b value must be 0, and the c value can be any positive or negative number. These requirements ensure that there is only one possible way to write an equation in vertex form.
For example, look at the equation Y = –6×2 + 3x + 2. In order to write this in vertex form, you must first identify the a (y-coordinate of vertex), b (x-coordinate of vertex), and c (magnitude of coordinate) values.
The a value is –2,thebvalueis0andthecvalueis2.
Plug a, b, and c values into y = a(x – h) + k format to find k
Now let’s go back to our original Y = –6×2 + 3x + 2 equation and see how we can rewrite it in vertex form.
First, remember that we can always rearrange equations as long as we do not change the values or order of operations. This makes it easy to rewrite an equation in vertex form!
So start by rearranging the equation into the following format: y = a(x − h) + k
Where h is the horizontal shift, a is the slope of the line, and k is the vertical shift of the line. Now, plug in some values for a, h, and k to find out what they are. Then check to see if the original equation equals this new one.
Use algebra to find k value
Now, let’s use our equation in vertex form to find the value of k. Since the variable k represents the height of the line, we can assume that k is equal to the y-coordinate of the vertex.
To find the value of k, we need to solve for it. To do this, we first have to isolate k on one side of the equation using algebra. Then we can solve for k by using math operations like addition, subtraction, multiplication, and division.
For example, if our equation in vertex form is y = –6×2 + 3x + 2, then isolating k on one side of the equation would look like this:
Then solving for k would look like this: k = 2 = 2×2 = 4k=4 Now that we have found the value of k, let’s plug it back into our original equation in vertex form:y=–6x²+3x+2.y=–6(4)²+3(4)+(2)(4)=16+12+8=36 Y=36!So now you know how to find both x and y values using vertex form.Happy mathming!With all of that under your belt, you are ready to solve more complex equations in vertex form!When an equation in slope-intercept form is transformed into an equation in vertex form, only one more variable is added. This makes it a bit easier to solve for x and y values separately.