# M:C:MC:20

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Learn more about Heart of Algebra.

If *k* is a positive constant different from 1, which of the following could be the graph of in the *xy*-plane?

Each of the four answer choices presents a graph in the *xy*-plane. The numbers negative 6 through 6 appear along both axes and the origin is labeled *O*.

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Choice B is correct. Manipulating the equation to solve for *y* gives revealing that the graph of the equation must be a line that passes through the origin. Of the choices given, only the graph shown in choice B satisfies these conditions.

Choice A is not the correct answer. The student may have seen that the term is a multiple of and wrongly concluded that this is the equation of a line with slope 1.

Choice C is not the correct answer. The student may have made incorrect steps when simplifying the equation or may have not seen the advantage that putting the equation in slope-intercept form would give in determining the graph, and thus wrongly concluded the graph has a nonzero *y*-intercept.

Choice D is not the correct answer. The student may not have seen that term can be multiplied out and the variables *x* and *y* isolated, and wrongly concluded that the graph of the equation cannot be a line.

Students must understand the relationship between an equation in two variables and the characteristics of its graph (for example, shape, position, intercepts, extreme points, or symmetry). In addition, students must transform the given equation into a more suitable form and then make the connection between the obtained equation and the graph.