Question

What is the vertex form of $y = 2{x^2} + 9x - 5$ ?

Answer 1

Hint: Here we are going to find the vertex form of the given quadratic function using the formula for vertex form of quadratic function.
Formula used:
The vector form of quadratic function is given by $f(x) = a{\left( {x - h} \right)^2} + k$ where $\left( {h,k} \right)$ is the vertex of the parabola.

Complete step by step solution:
To convert a quadratic form $y = a{x^2} + bx + c$ to vertex form$y = a{\left( {x - h} \right)^2} + k$ , we are using the process of completing the square,
Given that $y = 2{x^2} + 9x - 5$.
Since we will be completing the square, we will isolate the ${x^2}$ and $x$ terms, so move $ - 5$ to the other side of the equal sign we get,
$y + 5 = 2{x^2} + 9x$
Now, we need a leading coefficient of $1$ for completing the square, so factor out the current leading coefficient of $2$ , we get,
$y + 5 = 2\left( {{x^2} + \dfrac{9}{2}x} \right)$
Get ready to create a perfect square trinomial. When we add a number to both sides, the number will be multiplied by $2$ on both sides of the equal sign, and then we need to find the perfect square trinomial. Take the half of the coefficient $x$ term inside the parentheses, square it, and place it in both sides of the equal sign then the number is, $\dfrac{{81}}{{16}}$ ,
$y + 5 + 2\left( {\dfrac{{81}}{{16}}} \right) = 2\left( {{x^2} + \dfrac{9}{2}x + {{\left( {\dfrac{9}{4}} \right)}^2}} \right)$
Simplify and convert the right side to a squared expression, we get,
$y + 5 + \dfrac{{81}}{8} = 2{\left( {x + \dfrac{9}{4}} \right)^2}$
$y + \dfrac{{40 + 81}}{8} = 2{\left( {x + \dfrac{9}{4}} \right)^2}$
$y + \dfrac{{121}}{8} = 2{\left( {x + \dfrac{9}{4}} \right)^2}$
Isolate the $y$ term so move $\dfrac{{121}}{8}$ to the other side of the equal sign, we get,
$y = 2{\left( {x + \dfrac{9}{4}} \right)^2} - \dfrac{{121}}{8}$
This is the vertex form of the equation vertex $\left( {h,k} \right) = \left( { - \dfrac{9}{4}, - \dfrac{{121}}{8}} \right)$ .

Note: In some cases, it may need to transform the equation into the exact vertex form of$y = a{\left( {x - h} \right)^2} + k$, showing a subtraction sign in the parentheses before the $h$ term, and the addition of the $k$ term.