Question

How do you graph the parabola \[y = {\left( {x + 4} \right)^2} + 2\] using vertex, intercepts and additional points?

Answer 1

Hint: Here in this question, we have to plot the graph of a given parabolic equation using the vertex, intercepts and additional points. To find these by comparing the given equation to the vertex form of a quadratic equation \[y = a{\left( {x - h} \right)^2} + k\] where \[\left( {h,k} \right)\] is the vertex of parabola, then the x-intercept is found by putting \[y = 0\;\] in the equation similarly y-intercept is found by putting \[x = 0\] in the equation and the additional points found by giving x values as 0, 1, 2,… to the parabolic equation we get simultaneously the y values.

Complete step-by-step answer:
Consider the equation of parabola
\[y = {\left( {x + 4} \right)^2} + 2\]
Compare to this equation with the vertex form of a quadratic equation is given by \[y = a{\left( {x - h} \right)^2} + k\] where \[\left( {h,k} \right)\] is the vertex of the parabola. The h represents the horizontal shift and k represents the vertical shift.
Here \[h = - 4\], \[k = 2\], \[a = 1\] Since \[a\] is positive, the parabola opens upward.
Therefore, the vertex of parabola \[V\left( {h,k} \right) = V\left( { - 4,2} \right)\]
The Axis of symmetry is \[x = h\] or \[x = - 4\].
Since, the parabola is upward & above x-axis hence it wouldn't intersect the x-axis. Now, to find y-intercept we set \[x = 0\] in given equation, as follows i.e.,
\[ \Rightarrow y = {(0 + 4)^2} + 2\]
\[ \Rightarrow y = 16 + 2\]
\[ \Rightarrow y = 8\]
Hence, the parabola intersects the y-axis at \[\left( {0,18} \right)\]
Now find the additional points by giving the x values as 0, 1, 2,… to the parabolic equation we get simultaneously the y values.
The additional points are:

\[x\]	\[ - 6\]	\[ - 5\]	\[ - 4\]	\[ - 3\]	\[ - 2\]
\[y = {\left( {x + 4} \right)^2} + 2\]	\[6\]	\[3\]	\[2\]	\[3\]	\[6\]
\[\left( {x,y} \right)\]	\[\left( { - 6,6} \right)\]	\[\left( { - 5,3} \right)\]	\[\left( { - 4,2} \right)\]	\[\left( { - 3,3} \right)\]	\[\left( { - 2,6} \right)\]

The graph of parabola \[y = {\left( {x + 4} \right)^2} + 2\] is:

Note: When we see the equation we can easily recognize the kind of graph we can obtain. Usually the equation will be in the form of \[y = a{\left( {x - h} \right)^2} + k\]. Hence by substituting the value of x we can determine the value of y. The graph is plotted x-axis versus y-axis. The graph is of the form 2D.