Question

How do you write the vertex form equation of the parabola $y = {x^2} + 4x - 7$?

Answer 1

Hint: In this question, we need to write the given equation of the parabola into vertex form. Note that we use the standard form of parabola $a{x^2} + bx + c$ to find the values of a, b and c. Then we convert the given equation to a vertex form of a parabola given by $a{(x + d)^2} + e$, where $d = \dfrac{b}{{2a}}$ and $e = c - \dfrac{{{b^2}}}{{4a}}$.
Substitute these values in the formula and then we obtain the required vertex form.

Complete step-by-step solution:
Given an equation of the form $y = {x^2} + 4x - 7$ …… (1)
The given equation is in the standard form of parabola.
We are asked to find the vertex form given in the equation (1).
Note that the above given equation is a quadratic equation.
The standard form of the parabola is given by the equation,
$a{x^2} + bx + c$ …… (2)
Now comparing the equations (1) and (2) we get,
$a = 1,$ $b = 4$ and $c = - 7$.
Now we find out the vertex form using these values.
The vertex form of a parabola is given by the equation,
$a{(x + d)^2} + e$ …… (3)
where, we have $d = \dfrac{b}{{2a}}$ and $e = c - \dfrac{{{b^2}}}{{4a}}$.
Now we find the value of d.
We have $d = \dfrac{b}{{2a}}$, substituting $b = 4$ and $a = 1$,we get,
$ \Rightarrow d = \dfrac{4}{{2 \times 1}}$
$ \Rightarrow d = \dfrac{4}{2}$
$ \Rightarrow d = 2$
Hence the value of d is 2.
Now we find the value of e.
We have $e = c - \dfrac{{{b^2}}}{{4a}}$,
Substituting $a = 1,$ $b = 4$ and $c = - 7$, we get,
$ \Rightarrow e = - 7 - \dfrac{{{{(4)}^2}}}{{4 \times 1}}$
$ \Rightarrow e = - 7 - \dfrac{{16}}{4}$
We know that $\dfrac{{16}}{4} = 4$.
Hence, we get,
$ \Rightarrow e = - 7 - 4$
$ \Rightarrow e = - 11$
Hence the value of e is -11.
Now writing in the vertex form by substituting the values of a, d and e in the equation (3), we get,
$a{(x + d)^2} + e$
$ \Rightarrow 1{(x + 2)^2} - 11$
Let us set this vertex form equal to y.
Then we have, $y = 1{(x + 2)^2} - 11$
Which can also be written as,
$y = 1{(x - ( - 2))^2} + ( - 11)$
Hence the vertex form of the equation of the parabola $y = {x^2} + 4x - 7$is given by $y = 1{(x - ( - 2))^2} + ( - 11)$.

Note: Students must know the equations representing standard form and vertex form of the parabola. Since they may get confused while solving such problems. This applies only if the given equation is a quadratic equation.
The standard form of the parabola is given by the equation,
$a{x^2} + bx + c$
The vertex form of a parabola is given by the equation,
$a{(x + d)^2} + e$
where, $d = \dfrac{b}{{2a}}$ and $e = c - \dfrac{{{b^2}}}{{4a}}$.