Question

How do you graph \[y={{x}^{2}}+4x+6\]?

Answer 1

Hint: In order to plot the graph for the above equation , we need to find the coordinates for the same. The above equation is a quadratic equation so we can either solve the equation for x and then we can obtain the coordinates by putting values in x and getting the value of y or directly obtain the coordinates by putting values in x and getting the value of y. The values so obtained are written in the form \[(x,y)\]which are coordinates.

Complete step by step answer:
According to the question we have to plot the graph of the given function.
We will start with factorisation of \[{{x}^{2}}+4x+6\] if it is possible, as it would greatly reduce the calculation required.
To check whether \[{{x}^{2}}+4x+6\] has real roots, we will use discriminant, D. Only if D>0, the quadratic equation has real roots else it has complex or imaginary roots.
The formula of discriminant we have is:
\[D={{b}^{2}}-4ac\]
Substituting the values of b, a and c, we get
\[D=16-24=-8<0\]
Therefore, the quadratic equation has no real roots and only imaginary roots.
So we will do it straightaway by putting values in x and then calculating for the value of y.
By looking at the function we have, we know that the given expression or function is of a parabola in the positive y-axis direction.
We will start putting values in x,
When \[x=0\]
Putting the value of x in y, we get
\[y={{x}^{2}}+4x+6\]
\[y=0+0+6=6\]
Coordinate is \[(0,6)\]

When \[x=-1\]
Putting the value of x in y, we get
\[y={{(-1)}^{2}}+4(-1)+6\]
\[y=1-4+6=3\]
Coordinate is \[(-1,3)\]

When \[x=1\]
Putting the value of x in y, we get
\[y={{(1)}^{2}}+4(1)+6\]
\[y=1+4+6=11\]
Coordinate is \[(1,11)\]

When \[x=-2\]
Putting the value of x in y, we get
\[y={{(-2)}^{2}}+4(-2)+6\]
\[y=4-8+6=2\]
Coordinate is \[(-2,2)\]

When \[x=2\]
Putting the value of x in y, we get
\[y={{(2)}^{2}}+4(2)+6\]
\[y=4+8+6=18\]
Coordinate is \[(2,18)\]

When \[x=-3\]
Putting the value of x in y, we get
\[y={{(-3)}^{2}}+4(-3)+6\]
\[y=9-12+6=3\]
Coordinate is \[(-3,3)\]

When \[x=-4\]
Putting the value of x in y, we get
\[y={{(-4)}^{2}}+4(-4)+6\]
\[y=16-16+6=6\]
Coordinate is \[(-4,6)\]
Now we will plot the graph based on the obtained coordinates which is as follows:

Note: Start finding values for abscissa(x-coordinate) and ordinate(y-coordinate) by taking each as zero one by one. While plotting the graph be careful with the coordinates marking them at the correct position. Since, the equation given to us had no real roots so, we directly put the values in x and obtained the value of y. We can always factorize the equation, if possible, as it can shorten our calculations.