M:N:MC:7

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The graph of $y equals open parentheses 2 x minus 4 close parentheses open parentheses x minus 4 close parentheses$ is a parabola in the xy-plane. In which of the following equivalent equations do the x- and y-coordinates of the vertex of the parabola appear as constants or coefficients?

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$y equals 2 x squared minus 12 x plus 16$

Correct Answer:

$y equals 2 x open parentheses x minus 6 close parentheses plus 16$

Correct Answer:

$y equals 2 open parentheses x minus 3 close parentheses squared plus open parentheses minus 2 close parentheses$

Correct Answer:

Yes

$y equals open parentheses x minus 2 close parentheses open parentheses 2 x minus 8 close parentheses$

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Choice C is correct. The equation $y equals open parentheses 2 x minus 4 close parentheses open parentheses x minus 4 close parentheses$ can be written in vertex form, $y equals a open parentheses x minus h close parentheses squared plus k comma$ to display the vertex, $open parentheses h comma space k close parentheses comma$ of the parabola. To put the equation in vertex form, first multiply: $open parentheses 2 x minus 4 close parentheses open parentheses x minus 4 close parentheses equals 2 x squared minus 8 x minus 4 x plus 16.$ Then add like terms, $2 x squared minus 8 x minus 4 x plus 16 equals 2 x squared minus 12 x plus 16.$ The next step is completing the square.

$y equals 2 x squared minus 12 x plus 16$
$y equals 2 open parentheses x squared minus 6 x close parentheses plus 16$	Isolate the $x squared$ term by factoring
$y equals 2 open parentheses x squared minus 6 x plus 9 minus 9 close parentheses plus 16$	Make a perfect square in the parentheses
$y equals 2 open parentheses x squared minus 6 x plus 9 close parentheses minus 18 plus 16$	Move the extra term out of the parentheses
$y equals 2 open parentheses x minus 3 close parentheses squared minus 18 plus 16$	Factor inside the parentheses
$y equals 2 open parentheses x minus 3 close parentheses squared minus 2$	Simplify the remaining terms

Therefore, the coordinates of the vertex are $open parentheses 3 comma minus 2 close parentheses comma$ which are revealed in (C). Since students are told that all of the equations are equivalent, simply knowing the form that displays the coordinates of the vertex will save them all of these steps—this is known as “seeing structure in the expression or equation.”

Choice A is not the correct answer. This answer displays the location of the y-value of the y-intercept of the graph $open parentheses 0 comma space 16 close parentheses$ as a constant.

Choice B is not the correct answer. This answer displays the location of the y-value of the y-intercept of the graph $open parentheses 0 comma space 16 close parentheses$ as a constant.

Choice D is not the correct answer. This answer displays the location of the x-value of one of the x-intercepts of the graph $open parentheses 2 comma space 0 close parentheses$ as a constant.

Question Difficulty:

medium

Objective:

Students must be able to see structure in expressions and equations and create a new form of an expression that reveals a specific property.

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Question 7 of 18

M:N:MC:7

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