Question
Question: How do you write the vertex form equation of the parabola \(y={{x}^{2}}-4x+3\) ?...
How do you write the vertex form equation of the parabola y=x2−4x+3 ?
Solution
To get the equation of parabola in vertex form, we will have to convert the given equation that is the general form of parabola as y=x2−4x+3 in to the vertex form of the parabola that is y=a(x−h)2+k by adding or subtracting any number that will gives a form of the formula (a+b)2=a2+2ab+b2 or (a−b)2=a2−2ab+b2 , will help us to get the vertex form of the parabola.
Complete step by step solution:
Here, we have the general form of the parabola that is given in the question that is a quadratic equation also as:
⇒y=x2−4x+3
Since, we need to convert this equation in the vertex form where we will have to get a square term also. So, for getting the square term, we can write the above equation as:
⇒y=x2−2×2×x+3
Since, the above equation does not form a square term. So, for getting the square term we will add 1 both sides of the equation as:
⇒y+1=x2−2×2×x+3+1
Now, we will solve the above equation where we will add 3 and 1 that will give 4 as a result of addition as:
⇒y+1=x2−2×2×x+4
Since, 4 is the square of 2 . So, we can write 22 in the place of 4 as:
⇒y+1=x2−2×2×x+22
Here, the right hand side of the above expression shows the expansion a2−2ab+b2 that is the formula for (a−b)2 . So, we can write (x−2)2 in the above equation for the expression x2−2×2×x+4 as:
⇒y+1=(x−2)2
Now, we will subtract 1 both sides of the above equation to get the vertex form of the given general equation of parabola as:
⇒y+1−1=(x−2)2−1
Here, equal like numbers will be cancel out in the left side of the above equation as:
⇒y=(x−2)2−1
Hence, the above equation is the vertex form of the parabola.
Note: Here, we will try to find out the same given general equation of the parabola from the obtained vertex from of the parabola to check whether the solution is correct or not in the following way:
So, the obtained vertex form of the parabola is:
⇒y=(x−2)2−1
Now, we will open the bracket and will use the formula (a−b)2=a2−2ab+b2 as:
⇒y=x2−2×2×x+22−1
Now, we will complete the multiplication process; get the value of square in the above equation as:
⇒y=x2−4x+4−1
Here, we will subtract 1 from 4 as:
⇒y=x2−4x+3
Since, we got the given general equation of the parabola. Hence, the solution is correct.