The graph of 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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Choice C is correct. The equation can be written in vertex form, to display the vertex, of the parabola. To put the equation in vertex form, first multiply: Then add like terms, The next step is completing the square.
|Isolate the term by factoring|
|Make a perfect square in the parentheses|
|Move the extra term out of the parentheses|
|Factor inside the parentheses|
|Simplify the remaining terms|
Therefore, the coordinates of the vertex are 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 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 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 as a constant.
Students must be able to see structure in expressions and equations and create a new form of an expression that reveals a specific property.