Question

Sketch the graph of $y = {(x - 2)^2} - 7$ and describe the transformation?

Answer 1

Hint: According to the question given in the question we have to determine the sketch of the graph of the given quadratic expression which is $y = {(x - 2)^2} - 7$ and describe the transformation. So, to determine the sketch of the graph we have to determine the points to plot for which first of all as we can see that the given function is in the form of $a{(x - h)^2} + k$ where h is the axis of symmetry, and k is the maximum and minimum value of the function and it is also known as the vertex of the parabola.
Now, we have to determine the roots/zeros of the given quadratic expression and y –axis intercept. For which we have to use the formula which is as mentioned below:

Formula used:
$ \Rightarrow {(a - b)^2} = {a^2} + {b^2} - 2ab.............(A)$
Now, with the help of the roots/zeroes obtained we have to determine the y-axis intercepts.
Formula used:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}................(B)$
Where, a is the coefficient of ${x^2}$and x is the coefficient of x and c is the constant term.

Complete step by step answer:
First of all as we can see that the given function is in the form of $a{(x - h)^2} + k$ where h is the axis of symmetry, and k is the maximum and minimum value of the function and it is also known as the vertex of the parabola.
Now, we have to determine the roots/zeros of the given quadratic expression and y –axis intercept. For which we have to use the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow {x^2} + 4 - 4x - 7 = 0 \\
   \Rightarrow {x^2} - 4x - 3 = 0 \\
 $

Now, with the help of the roots/zeroes obtained we have to determine the y-axis intercepts. Hence, to determine the roots of the quadratic expression as obtained in the solution step 2 we have to use the formula (B) which is as mentioned in the solution hint.
$
   \Rightarrow x = \dfrac{{ - ( - 4) \pm \sqrt {{{( - 4)}^2} - 4 \times 1 \times ( - 3)} }}{{2 \times 1}} \\
   \Rightarrow x = \dfrac{{4 \pm \sqrt {16 + 12} }}{2} \\
 $
On solving the expression as obtained just above,
$ \Rightarrow x = \dfrac{{4 \pm 2\sqrt 7 }}{2}$
Eliminating 2 in the expression as obtained just above,
$ \Rightarrow x = 2 \pm \sqrt 7 $

With the help of the quadratic formula (B) we have determined the roots which will satisfy the quadratic expression and will make it 0. Hence, roots are
$ \Rightarrow (2 + \sqrt 7 ,0)$and$(2 - \sqrt 7 ,0)$,
y-axis intercept is where, x = 0.
$
   \Rightarrow y = {(0)^2} - 4(0) - 3 \\
   \Rightarrow y = - 3 \\
 $
$\therefore$ Points obtained are:
\[ \Rightarrow (0, - 3)\]
We have obtained all the plotting points which we have to plot in the quadrant plane are $( - 2,7),(2 + \sqrt 7 ,0),(2 - \sqrt 7 ,0),(0, - 3)$

Hence, with the help of formula (A) and (B) we have determined the graph for the given expression and this can be viewed as the graph of $y = {x^2}$ which is 2 units to the right and 7 units in the –y direction.

Note:
• The given function is in the form of $a{(x - h)^2} + k$ where h is the axis of symmetry and k is the maximum or minimum value of the function and this is also known as the vertex of the parabola.
• To determine the roots/zeros of the given quadratic expression and y –axis intercept we have to use the quadrant formula but before that we have to determine the values of a, b, and c which are the coefficient of ${x^2}$, coefficient of x and the constant terms respectively.