The function in vertex form is equivalent to
.
<h3>What are quadratic equations?</h3>
A quadratic equation is an equation of degree 2 in other words.Quadratic equations have the form ax² + bx + c = 0 and are second-degree algebraic expressions.
The standard quadratic function with vertex (h, k) is ;
![\rm y = a(x - h)^2 + k](https://tex.z-dn.net/?f=%5Crm%20y%20%3D%20a%28x%20-%20h%29%5E2%20%2B%20k)
The general quadratic equation is found as;
![\rm y = ax^2 + bx + c](https://tex.z-dn.net/?f=%5Crm%20y%20%3D%20ax%5E2%20%2B%20bx%20%2B%20c)
The value of h is given as;
![\rm h = \frac{-b}{2a}](https://tex.z-dn.net/?f=%5Crm%20h%20%3D%20%5Cfrac%7B-b%7D%7B2a%7D)
For the given equation in the problem;
![\rm y = x^2 + x + 1](https://tex.z-dn.net/?f=%5Crm%20y%20%3D%20x%5E2%20%2B%20x%20%2B%201)
The value of the k is found by;
![\rm y =( \frac{-1}{2}) ^2 - \frac{1}{2} + 1 \\\\ y= \frac{1}{4} - \frac{1}{2}+ 1 \\\\ y= \frac{1}{4}- \frac{2}{4}+ \frac{4}{4}\\\\ y= \frac{3}{4}](https://tex.z-dn.net/?f=%5Crm%20y%20%3D%28%20%5Cfrac%7B-1%7D%7B2%7D%29%20%5E2%20-%20%5Cfrac%7B1%7D%7B2%7D%20%20%2B%201%20%5C%5C%5C%5C%20y%3D%20%5Cfrac%7B1%7D%7B4%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%2B%201%20%5C%5C%5C%5C%20y%3D%20%5Cfrac%7B1%7D%7B4%7D-%20%5Cfrac%7B2%7D%7B4%7D%2B%20%5Cfrac%7B4%7D%7B4%7D%5C%5C%5C%5C%20y%3D%20%5Cfrac%7B3%7D%7B4%7D)
So the vertex is at
and
then the vertex form is:
![y = (x +\frac{2}{4} )^2 + \frac{3}{4}](https://tex.z-dn.net/?f=y%20%3D%20%28x%20%2B%5Cfrac%7B2%7D%7B4%7D%20%29%5E2%20%2B%20%5Cfrac%7B3%7D%7B4%7D)
Hence the functions in vertex form are equivalent
.
To learn more about the quadratic functions refer to the link;
brainly.com/question/1214333