Sometimes you can add useful lines, points or circles to a geometric figure to facilitate solving a problem. You can also add any given information — as well as any new information as you derive it — to the figure to help you see relationships within the figure more easily, for example, the length of a line segment or the measure of an angle.

• This strategy is used in the following two sample questions.

**This is a Multiple-Choice – Select One Answer Choice Question.**

The figure above shows the graph of the function

*f*, defined by for all numbers*x*. For which of the following functions*g,*defined for all numbers*x,*does the graph of*g*intersect the graph of*f*?(A)

(B)

(C)

(D)

(E)*Explanation*You can see that all five choices are linear functions whose graphs are lines with various slopes and

*y*-intercepts. The graph of Choice A is a line with slope 1 and*y*-intercept shown in the following figure.It is clear that this line will not intersect the graph of

*f*to the left of the*y*-axis. To the right of the*y*-axis, the graph of*f*is a line with slope 2, which is greater than slope 1. Consequently, as the value of*x*increases, the value of*y*increases faster for*f*than for*g*, and therefore the graphs do not intersect to the right of the*y*-axis. Choice B is similarly ruled out. Note that if the*y*-intercept of either of the lines in choices A and B were greater than or equal to 4 instead of less than 4, they would intersect the graph of*f*.Choices C and D are lines with slope 2 and

*y*-intercepts less than 4. Hence, they are parallel to the graph of*f*(to the right of the*y*-axis) and therefore will not intersect it. Any line with a slope greater than 2 and a*y*-intercept less than 4, like the line in Choice E, will intersect the graph of*f*(to the right of the*y*-axis).**The correct answer is Choice E**,**Note:**This question and explanation also appear as an example of Strategy 3.

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