Question

How do you find the vertex of $y = {x^2} + 4x + 2$ ?

Answer 1

Hint: The given quadratic equation belongs to the family of parabola. To find the vertex of the given quadratic equation we will change it into the equation of vertex form. We know that the equation of parabola in vertex form is $y = a{\left( {x - h} \right)^2} + k$ where $\left( {h,k} \right)$ represents the vertex of the parabola. The given equation of parabola does not pass through the origin because it is not satisfying the coordinate of the origin. To find the vertex of the given parabola, we will convert it into a standard form of parabola. Then we will compare the given equation with the standard form of equation to find the coefficients of the equation. After comparing the equation we will substitute the value in the formula of vertex.

Complete step by step solution:
Step: 1 the given quadratic equation is,
$y = {x^2} + 4x + 2$
Compare the given equation with the standard equation of quadratic to find the coefficients.
The standard form of equation of the quadratic is,
$y = a{x^2} + bx + c$
Therefore,
$
   \Rightarrow a = 1 \\
   \Rightarrow b = 4 \\
   \Rightarrow c = 2 \\
 $
We know that the vortex $\left( {h,k} \right)$ of the equation $y = a{x^2} + bx + c$ is given by the formula
$
  h = \dfrac{{ - b}}{{2a}} \\
  k = f\left( h \right) \\
 $
Step: 2 to find the value of the vortex, substitute the value of the coefficients if the given formula.
$
   \Rightarrow h = \dfrac{{ - b}}{{2a}} \\
   \Rightarrow h = \dfrac{{ - 4}}{{2 \times 1}} \\
   \Rightarrow h = - 2 \\
 $
To find the value of the vortex $k$ , substitute the value of $h$ in the equation.
Therefore
$k = - {2^2} + 4 \times \left( { - 2} \right) + 1$
Now solve the equation to find the value.
$
   \Rightarrow k = 4 - 8 + 1 \\
   \Rightarrow k = - 3 \\
 $
Final Answer:
Therefore the vortex of the given quadratic equation $y = {x^2} + 4x + 2$ is $\left( { - 2, - 3} \right)$.

Note:
Students are advised to not make any mistake, while writing the value of $k$ as it makes confusion. They must remember here that $y$ is a function of $x$. They should compare the given quadratic equation with the standard form of the equation.