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Question 27 of 30

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If k is a positive constant different from 1, which of the following could be the graph of y minus x equals k left parenthesis x space plus space y right parenthesis 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.

Select an Answer

Choice A. The graph shows a line that goes up from left to right and intersects the x-axis at negative 2, and the y-axis at 2.

Correct Answer: 
No

Choice B. The graph shows a line that goes down from left to right and goes through the origin. The line appears to pass through the point with coordinates negative one comma 2.

Correct Answer: 
Yes

Choice C. The graph shows a line that goes up from left to right and intersects the y-axis at negative 3, and the x-axis at one point 5.

Correct Answer: 
No

Choice D. The graph shows a smooth curve that appears to be a parabola. The parabola has its vertex at the origin, opens up and passes through the point with coordinates one comma one.

Correct Answer: 
No

View Correct Answer

Choice B is correct. Manipulating the equation to solve for y gives y equals left parenthesis fraction numerator 1 plus k over denominator 1 minus k end fraction right parenthesis x comma 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 k left parenthesis x plus y right parenthesis is a multiple of x plus y 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 k left parenthesis x plus y right parenthesis can be multiplied out and the variables x and y isolated, and wrongly concluded that the graph of the equation cannot be a line.

Question Difficulty: 
hard
Objective: 

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.