Question

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.

\({\bf{y = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 6x + 4}}\)

Short answer

The graph is shown below.

Step-by-step solution

Given data

We are given the transformed graph as\(y = {x^2} + 6x + 4\),

Solution \(\)\(\) \(\)

We can put this form of a “completing square” learned in algebra.

First, put a space in-between the 2^nd and 3^rd terms,

\(y = {x^2} + 6x\;\,\;\; + 4\)

Second, take\(\frac{1}{2}\)of middle coefficient, square it, and add and subtract from the right side,

\(y = {x^2} + 6x + {3^2} + \;\;\;\;4 - {3^2}\)

Third, the first three terms become the square of a binomial\({\left( {x + 3} \right)^2}\),

hence\(y = {\left( {x + 3} \right)^2} - 5\).

The transformed graph is given in purple below.

We can now see that the basic graph is given by\(f\left( x \right) = {x^2}\)as seen in red below.

Next the \( - 5\) shifts the blue graph by 5 units down. The final transformed graph has the colour purple.

The graph is given below