Question

How do you graph \[f\left( x \right) = {\left( {x + 2} \right)^2}?\]

Answer 1

Hint: The given question describes the operation of addition/ subtraction/ multiplication/ division. Also, this problem involves the operation of substituting the \[x\] values in the given equation to find \[y\] values. Also, \[y\] is the fraction of \[x\]. By using the values of \[x\] and \[y\] we can easily draw the graph. To make easy calculations we can simplify the given equation by using algebraic formulas.

Complete step by step solution:
The given question is shown below,
\[y = f\left( x \right) = {\left( {x + 2} \right)^2} \to \left( 1 \right)\]
We know that,
\[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
By using the above-mentioned algebraic formula we can simplify the equation \[\left( 1 \right)\] as follows,
\[\left( 1 \right) \to y = {\left( {x + 2} \right)^2}\]
\[
y = {x^2} + 2 \times 2 \times x + {2^2} \\
y = {x^2} + 4x + 4 \to \left( 2 \right) \\
\]
We would draw the graph for the above equation.
As a first step, we would assume \[x\] value as given below,
\[x = ... - 2, - 1,0,1,2,.....\]
By substituting the above-mentioned \[x\] values in the equation \[\left( 2 \right)\], we can find the \[y\] values.
Let’s substitute \[x = - 2\] in the equation \[\left( 2 \right)\], we get
\[\left( 2 \right) \to y = {x^2} + 4x + 4\]
\[
y = {\left( { - 2} \right)^2} + \left( {4 \times - 2} \right) + 4 \\
y = 4 - 8 + 4 \\
y = 0 \\
\]
Let’s substitute\[x = - 1\] in the equation \[\left( 2 \right)\], we get
\[\left( 2 \right) \to y = {x^2} + 4x + 4\]
\[
y = {\left( { - 1} \right)^2} + 4\left( { - 1} \right) + 4 \\
y = 1 - 4 + 4 \\
y = 1 \\
\]
Let’s substitute \[x = 0\] in the equation \[\left( 2 \right)\], we get
\[\left( 2 \right) \to y = {x^2} + 4x + 4\]
\[
y = {\left( 0 \right)^2} + 4\left( 0 \right) + 4 \\
y = 4 \\
\]
Let’s substitute \[x = 1\] in the equation \[\left( 2 \right)\], we get
\[\left( 2 \right) \to y = {x^2} + 4x + 4\]
\[
y = {\left( 1 \right)^2} + 4\left( 1 \right) + 4 \\
y = 9 \\
\]
Let’s substitute \[x = 2\] in the equation \[\left( 2 \right)\], we get
\[\left( 2 \right) \to y = {x^2} + 4x + 4\]
\[
y = {\left( 2 \right)^2} + \left( {4 \times 2} \right) + 2 \\
y = 4 + 8 + 2 \\
y = 14 \\
\]
Let’s make a tabular column by using the\[x\]and\[y\]values as given below,

\[x\]	\[ - 2\]	\[ - 1\]	\[0\]	\[1\]	\[2\]
\[y\]	\[0\]	\[1\]	\[4\]	\[9\]	\[14\]

By using these points we can easily draw the graph.

The above graph represents the equation \[y\left( x \right) = {\left( {x + 2} \right)^2}\]

Note: In this type of question we would assume \[x\] value, by using the \[x\] value we can find the value of \[y\]. The graph could be based on the equation. \[y\] is the function of \[x\]. So, \[y\] also can be written as \[f\left( x \right)\]. Note that \[y = {x^2}\] the form equation always makes a parabolic shape in the graph sheet. Remember the algebraic formula \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\].