Question

The graph of the given equation \[y = 2{x^2} + 4x + 3\] has its
A) Has its lowest point(-1,9)
A. Has its lowest point(1,1)
B. Has its lowest point at (-1,1)
C. Has its highest point at (-1,9)
D. Has its highest point at (-1,1)

Answer 1

Hint: The given questions asks to find the either the highest or the lowest point of the given equation. In order to do that, we will at first figure out the shape of the equation. As we can see that it is a quadratic equation, where coefficient of x is not zero, thus it is safe that the equation is of parabola. And then to To find the lowest or the highest point of the parabola we find the vertex of the parabola.

Complete step by step answer:
Firstly let us learn about parabolas before dwelling into the question.
When you see a quadratic equation of form \[y = a{x^2} + bx + c\], where \[a \ne 0\] then it is safe to say that the equation is of parabola. It is said that the “standard form” of a parabola is a quadratic equation.
Now see, if the equation is of format \[y = a{(x - h)^2} + k\]then the parabola opens vertically.
And if the equation is of the form \[x = a{(y - k)^2} + h\]then the parabola opens horizontally.
now, talking about the question in hand,
we have given equation,
\[y = 2{x^2} + 4x + 3\]
We can see that the equation is of the first format, therefore, the given parabola opens vertically.
The vertex of the parabola is given by, \[x = \dfrac{{ - b}}{{2a}}\]where b is the coefficient of x and a is the coefficient of x squared.
Therefore,
\[ \Rightarrow x = \dfrac{{ - 4}}{{2(2)}} = - 1\]
If \[x = - 1\]then putting the value of x in given equation we get,
\[y = 2{( - 1)^2} + 4( - 1) + 3\]
\[ \Rightarrow y = 2 - 4 + 3\]
\[ \Rightarrow y = 1\]
Therefore, the parabola has its lowest point at, \[( - 1,1)\]
whose graph is given below,

So, the correct answer is Option B.

Note: The equations \[y = {x^2} + 3x - 10\] and \[x = 2{y^2} - 3y + 10\] are both parabola. You see an x squared but no y squared in the first equation, and you see a y squared but no x squared in the second equation. But nothing matters—signs and coefficients influence the overall appearance of the parabola (how it opens or how fat it is) but do not alter the fact that it is a parabola.