Question

Graph the following equation $ y = 2{x^2} + 12x + 16 $

Answer 1

Hint: In this problem, we have to graph the following equation $ y = 2{x^2} + 12x + 16 $ and this equation is similar to the general equation of parabola i.e, $ y = a{x^2} + bx + c $ . A plane curve that is almost u-shaped or we can say is of a bowl shaped and symmetrical in nature is called a parabola.
Formula Used: The formula of the axis of symmetry(x) is,
 $ x = \dfrac{{ - b}}{{2a}} $
Where, b and a is used from the general equation of parabola i.e, $ y = a{x^2} + bx + c $ .

Complete step by step solution:
Firstly, we will compare the general equation of the parabola and the equation given in the question. On comparing, we get,
 $ \Rightarrow a = 2,b = 12,c = 16 $
Secondly, we need to find the axis of symmetry by the formula given above.
 $
   \Rightarrow x = \dfrac{{ - 12}}{{2 \times 2}} \\
   \Rightarrow x = \dfrac{{ - 12}}{4} \\
   \Rightarrow x = - 3 \;
  $
Now, $ - 3 $ is the axis of symmetry and now, we will find the value of the vertex and it is found, when we substitute the value of (x) axis of symmetry in the equation given in the question.
 $
   \Rightarrow y = 2{\left( { - 3} \right)^2} + 12\left( { - 3} \right) + 16 \\
   \Rightarrow y = 2\left( 9 \right) + 12 \times \left( { - 3} \right) + 16 \\
   \Rightarrow y = 2 \times 9 - 36 + 16 \\
   \Rightarrow y = 18 - 20 \\
   \Rightarrow y = - 2 \;
  $
Now, $ - 2 $ is the vertex and now we have our first point to locate on the graph i.e, $ \left( { - 3, - 2} \right) $ .
Now, to find the other coordinates, different values of x are placed in the equation and then the values of y are computed and then, we will form a table of values through which we can plot a graph.
For example, if the value of x is $ - 5 $ , then, the value of y is $ 6 $ .
 $
   \Rightarrow y = 2{\left( { - 5} \right)^2} + 12\left( { - 5} \right) + 16 \\
   \Rightarrow y = 2 \times 25 + 12 \times \left( { - 5} \right) + 16 \\
   \Rightarrow y = 50 - 60 + 16 \\
   \Rightarrow y = - 10 + 16 \\
   \Rightarrow y = 6 \;
  $
Now, we will form a table for the values of x and y.

Value of x	Value of y
$ - 5 $	$ 6 $
$ - 4 $	$ 0 $
$ - 3 $	$ - 2 $
$ - 2 $	$ 0 $
$ - 1 $	$ 6 $

Hence, the graph formed by these points is,

Note: To make the graph of the following equation $ y = 2{x^2} + 12x + 16 $ , firstly, we need to find the axis of symmetry and the vertex and after that we will assume the values of x and substitute in the equation to find the value of y, and after finding the values we will locate these points on the graph.