Question

How do you graph the parabola \[y = - \dfrac{1}{4}{x^2}\] using vertex intercepts?

Answer 1

Hint: Here in this question, we have to plot the graph of 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
\[y = - \dfrac{1}{4}{x^2}\]
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 \[h = 0\], \[k = 0\], \[a = - 1\] Since \[a\] is negative, the parabola opens downward.
Therefore, the vertex of parabola \[V\left( {h,k} \right) = V\left( {0,0} \right)\]it means the parabola initiates at the origin and x intercepts is also at \[\left( {0,0} \right)\]
The Axis of symmetry is \[x = h\] or \[x = 0\].
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.,
\[ \Rightarrow y = - \dfrac{1}{4}{\left( 0 \right)^2}\]
\[ \Rightarrow y = 0\]
Hence, the parabola intersects the y-axis at \[\left( {0,0} \right)\]
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:

\[x\]	\[ - 2\]	\[ - 1\]	\[0\]	\[1\]	\[2\]
\[y = - \dfrac{1}{4}{x^2}\]	\[ - 1\]	\[ - \dfrac{1}{4}\]	\[0\]	\[ - \dfrac{1}{4}\]	\[ - 1\]
\[\left( {x,y} \right)\]	\[\left( { - 2, - 1} \right)\]	\[\left( { - 1, - \dfrac{1}{4}} \right)\]	\[\left( {0,0} \right)\]	\[\left( {1, - \dfrac{1}{4}} \right)\]	\[\left( {2, - 1} \right)\]

The graph of parabola \[y = - \dfrac{1}{4}{x^2}\] 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.