Question

How do you graph the parabola \[y = {\left( {x - 2} \right)^2} - 3\] 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
\[y = {\left( {x - 2} \right)^2} - 3\]
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 \[h = 2\], \[k = - 3\], \[a = 1\] Since \[a\] is positive, the parabola opens upward.
Therefore, the vertex of parabola \[V\left( {h,k} \right) = V\left( {2, - 3} \right)\]
The Axis of symmetry is \[x = h\] or \[x = 2\].
The y-intercept is found by putting \[x = 0\] in the equation i.e.,
\[ \Rightarrow y = {\left( {0 - 2} \right)^2} - 3\]
\[ \Rightarrow y = 4 - 3\]
\[ \Rightarrow y = 1\]
Hence, the parabola intersects the y-axis at \[\left( {0,1} \right)\]
Similarly, the x-intercept is found by putting \[y = 0\] in the equation i.e.,
\[ \Rightarrow 0 = {\left( {x - 2} \right)^2} - 3\]
On rearranging
\[ \Rightarrow {\left( {x - 2} \right)^2} = 3\]
On simplification, we het
\[ \Rightarrow x - 2 = \pm \sqrt 3 \]
\[ \Rightarrow x = 2 \pm \sqrt 3 \]
\[ \Rightarrow x = 2 + \sqrt 3 \] and \[x = 2 - \sqrt 3 \]
\[ \Rightarrow x \cong 3.732\] and \[x \cong 0.268\]
Hence, the parabola intersects the x-axis at \[\left( {3.732,0} \right)\] and \[\,\left( {0.268,0} \right)\]
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:

\[x\]	\[0\]	\[1\]	\[2\]	\[3\]	\[4\]
\[y = {\left( {x - 2} \right)^2} - 3\]	\[1\]	\[ - 2\]	\[ - 3\]	\[ - 2\]	\[1\]
\[\left( {x,y} \right)\]	\[\left( {0,1} \right)\]	\[\left( {1, - 2} \right)\]	\[\left( {2, - 3} \right)\]	\[\left( {3, - 2} \right)\]	\[\left( {4,1} \right)\]

The graph of parabola \[y = {\left( {x - 2} \right)^2} - 3\] 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