Example 6
Completion requirements
| Example 6 |
Jenna is asked to convert \(f\left( x \right) = 2x^2 + 4x + 8\) into vertex form, and then verify her solution. Her work is shown.
\(\begin{array}{l}
f\left( x \right) = 2x^2 + 4x + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 1 \\
f\left( x \right) = 2\left( {x + 1} \right)^2 + 7 \end{array}\)
Jenna now verifies her work by expanding the vertex form, and comparing the result to the original function.
Noticing the two are not the same, Jenna decides to use technology to see if there is a difference in the graphs. She enters the two equations into Y1 and Y2 , and can see from the second screen capture that there are two different graphs; she realizes an error must have been made.
Where is Jenna's error?
\(\begin{array}{l}
f\left( x \right) = 2x^2 + 4x + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 1 \\
f\left( x \right) = 2\left( {x + 1} \right)^2 + 7 \end{array}\)
Jenna now verifies her work by expanding the vertex form, and comparing the result to the original function.
| Expansion and simplification of vertex form |
Standard form
|
|
\(\begin{array}{l}
f\left( x \right) = 2\left( {x + 1} \right)^2 + 7 \\ f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 7 \\ f\left( x \right) = 2x^2 + 4x + 2 + 7 \\ f\left( x \right) = 2x^2 + 4x + 9 \end{array}\) |
\(f\left( x \right) = 2x^2 + 4x + 8\)
|
Noticing the two are not the same, Jenna decides to use technology to see if there is a difference in the graphs. She enters the two equations into Y1 and Y2 , and can see from the second screen capture that there are two different graphs; she realizes an error must have been made.
Where is Jenna's error?
Although the second step in the completion of the square is correct, the third step needs to be looked at carefully. Mistakes are frequently made on this step.
\(f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 1\)
Jenna is correct in adding 1 to make the perfect square trinomial, but the problem lies in subtracting 1 outside the brackets. Although she added 1 inside the bracket, the bracket is multiplied by 2. This means that she has actually added 2 to the function because of the distributive property. She needs to subtract 2 in this step instead of subtracting 1.
This is why it is strongly suggested to both add and subtract the constant first, then remove the subtracted value out of the brackets by multiplying it by a.
\(\begin{array}{l}
f\left( x \right) = 2x^2 + 4x + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1 - 1} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 2\left( 1 \right) \\
f\left( x \right) = 2\left( {x + 1} \right)^2 + 6 \end{array}\)
Check by expanding and simplifying.
Because the two functions are the same, the conversion was completed correctly.
Verification using a calculator produces the following screen captures. Notice there is only one graph, which indicates that Jenna has converted her function correctly from one form to another.
\(f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 1\)
Jenna is correct in adding 1 to make the perfect square trinomial, but the problem lies in subtracting 1 outside the brackets. Although she added 1 inside the bracket, the bracket is multiplied by 2. This means that she has actually added 2 to the function because of the distributive property. She needs to subtract 2 in this step instead of subtracting 1.
This is why it is strongly suggested to both add and subtract the constant first, then remove the subtracted value out of the brackets by multiplying it by a.
\(\begin{array}{l}
f\left( x \right) = 2x^2 + 4x + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1 - 1} \right) + 8 \\
f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 8 - 2\left( 1 \right) \\
f\left( x \right) = 2\left( {x + 1} \right)^2 + 6 \end{array}\)
Check by expanding and simplifying.
| Expansion and simplification of vertex form |
Standard form
|
| \(\begin{array}{l} f\left( x \right) = 2\left( {x + 1} \right)^2 + 6 \\ f\left( x \right) = 2\left( {x^2 + 2x + 1} \right) + 6 \\ f\left( x \right) = 2x^2 + 4x + 2 + 6 \\ f\left( x \right) = 2x^2 + 4x + 8 \end{array}\) |
\(f\left( x \right) = 2x^2 + 4x + 8\)
|
Because the two functions are the same, the conversion was completed correctly.
Verification using a calculator produces the following screen captures. Notice there is only one graph, which indicates that Jenna has converted her function correctly from one form to another.