Question

How do you sketch the graph of \[y = {\left( {x - 3} \right)^2}\] and describe the transformation?

Answer 1

Hint: Here we have to show the graphical representation of \[y = {\left( {x - 3} \right)^2}\] . To do so, our first approach is to find different values of \[y\] for the different values of \[x\] . Then, we will plot the obtained points on a graph. Therefore we can sketch the graph of \[y = {\left( {x - 3} \right)^2}\] by joining all the points with a curved line. Also, in the given equation \[y = {\left( {x - 3} \right)^2}\] , the R.H.S. is whole squared which means that \[y\] will always be a non-negative number. Therefore, for all values of \[x\] , the graph will lie above the x-axis, i.e. in the first and second quadrant.

Complete step by step answer:
Given equation is \[y = {\left( {x - 3} \right)^2}\] .
Now we have to find different coordinates which satisfy the equation, i.e. those points which lie on the above curve.
Hence, we will put random values for \[x\] and find the related value of \[y\] for that \[x\] or vice-versa.
Now, putting the value \[x = 0\] in the equation \[y = {\left( {x - 3} \right)^2}\] ,
We get,
\[y = {\left( {0 - 3} \right)^2}\]
\[y = {\left( { - 3} \right)^2}\]
Solving the R.H.S., we get
\[y = 9\]
Therefore, \[y = 9\] for \[x = 0\] .
Similarly, putting the value \[y = 0\] in \[y = {\left( {x - 3} \right)^2}\]
We get,
\[0 = {\left( {x - 3} \right)^2}\]
Taking square root of both sides,
\[x - 3 = 0\]
Adding 3 both sides, we get
\[x = 3\]
Therefore, \[x = 3\] for \[y = 0\]
Again, putting the value \[x = 4\] in the equation \[y = {\left( {x - 3} \right)^2}\]
We get,
\[y = {\left( {4 - 3} \right)^2}\]
Solving the R.H.S., we get
\[y = {\left( 1 \right)^2}\]
or
\[y = 1\]
Therefore, \[y = 1\] for \[x = 4\]
Now the three obtained points are:

\[x\]	\[0\]	\[3\]	\[4\]
\[y\]	\[9\]	\[0\]	\[1\]

Therefore, the three coordinates that lie on the curve \[y = {\left( {x - 3} \right)^2}\] are \[\left( {0,9} \right)\] , \[\left( {3,0} \right)\] and \[\left( {4,1} \right)\] .
Plotting these points on a graph and then extending the curve after joining all the points gives us the graph of \[y = {\left( {x - 3} \right)^2}\] .
The obtained graph is shown below:

That is the required graphical representation of the equation \[y = {\left( {x - 3} \right)^2}\] .
When the equation is given in its particular form as \[y = {\left( {x - a} \right)^2}\] , then the curve is shifted by \[a\] units on the x-axis. In that case, the point of contact of the curve and the x-axis is \[\left( {a,0} \right)\] . For example, in the above equation \[y = {\left( {x - 3} \right)^2}\] , the curve was shifted by \[3\] units on the x-axis.

Note:
The standard form of such curves is \[y = {x^2}\] . That curve also looks similar to the curve in the above graph but the curve touches the x-axis at the origin \[\left( {0,0} \right)\] . The given graph depicts an upward facing parabola touching the x axis at $\left( {3,0} \right)$.