Question

How do you find the vertex of $f\left( x \right) = 2{x^2} - 4x + 6$?

Answer 1

Hint: The equation for vertex form is given as: $y = a{(x - h)^2} + k$. The given equation in the question is in the quadratic form of $y = a{x^2} + bx + c$ , thus we have to change the quadratic form into the vertex form. In order to do so, first we have to find out the coordinates of the vertex and then just substitute the values in the equation.

Formula used: $y = a{(x - h)^2} + k$ , $y = a{x^2} + bx + c$, $ - \dfrac{b}{{2a}}$

Complete step-by-step solution:
The given equation is: $y = 2{x^2} - 4x + 6$, this equation is in the form of the quadratic equation: $y = a{x^2} + bx + c$, thus we can find the values of the variable $a{\text{ and }}b$
Comparing both the equations, we get:
$a = 2$
$b = - 4$
$c = 6$
Now we need to find the coordinates of the vertex which is given as $(h,k)$ where $h$ represents the x coordinate and $k$ represents the y coordinates
The formula to find $h$ coordinate of the vertex is given as: $ - \dfrac{b}{{2a}}$
Substituting the values of $b\;and\;a$ , we get:
\[ \Rightarrow h = - \dfrac{b}{{2a}} = - \dfrac{{\left( { - 4} \right)}}{{2 \times 2}} = \dfrac{4}{4} = 1\]
Now, in order to find the value of $k$, we substitute the known values in $y = 2{x^2} - 4x + 6$
Here $x$ is represented by $h$ coordinate and $y$ is represented by $k$, thus:
\[ \Rightarrow k = 2{x^2} - 4h + 6\]
On substituting the values of $h$, we get:
$ \Rightarrow k = 2{\left( 1 \right)^2} - 4\left( 1 \right) + 6$
On further simplifying, we get:
$ \Rightarrow k = 2 - 4 + 6$
On further simplifying we get:
$ \Rightarrow k = 4$
Therefore, $h = 1{\text{ and }}k = 4$
Putting these values in the vertex form equation $y = a{(x - h)^2} + k$ , we get:
$ \Rightarrow y = a{(x - h)^2} + k$
\[ \Rightarrow y = 2{\left( {x - \left( 1 \right)} \right)^2} + 4\], where $a = 2$ , $h = 1$ and $k = 4$
On further simplifying, we get:
$ \Rightarrow y = 2{\left( {x - 1} \right)^2} + 4$

$y = 2{\left( {x - 1} \right)^2} + 4$ is the required answer.

Note: The graph of a quadratic function is a parabola. The vertex is nothing but the highest or lowest point of a parabola, depending if its downward shaped or upward shaped respectively. The shape of the parabola is determined by the coefficient $a$ of the quadratic function.
If $a > 0$, then the graph makes a smile, which means it is an upward graph.
If $a < 0$, then the graph makes a frown, which means it is a downward graph