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Strategy 6: Add to a Geometric Figure

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 sample question.

This is a Multiple-Choice – Select One Answer Choice question.


  1. The figure shows the graph in the x y plane of the function f of x equals the absolute value of 2x, end absolute value, + 4. There are equally spaced tick marks along the x axis and along the y axis. The first tick mark to the right of the origin, and the first tick mark above the origin, are both labeled 1. The graph of the function f is in the shape of the letter V. It is above the x axis and is symmetric with respect to the y axis. The lowest point on the graph of f is the point 0 comma 4, which is located on y axis at the fourth tick mark above the origin. Going leftward from the point 0 comma 4, the graph of f is a line that slants upward, passing through the point negative 2 comma 8. Going rightward from the point 0 comma 4, the graph of f is a line that slants upward, passing through the point 2 comma 8.

    The figure above shows the graph of the function f, defined by f of x = the absolute value of 2x, end absolute value, + 4for 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) g of x = x minus 2
    (B) g of x = x + 3
    (C) g of x = 2x minus 2
    (D) g of x = 2x + 3
    (E) g of x = 3x minus 2

     

    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 Negative 2 shown in the following figure.

    Figure 6 is the same as figure 5 except that the graph of the line with slope 1 and y intercept negative 2 has been added. The line slants upward as you go from left to right and intersects the x axis at 2. The line is below the graph of y equals f of x.

    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, g of x = 3x minus 2

    Note: This question and explanation also appear as an example of Strategy 3.

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