General Problem-Solving Steps

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.

Sample Question 1 for Strategy 6: Multiple-Choice – Select One Answer Choice Question.

Begin figure description.

The figure shows the graph in the xy-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 the 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.

End figure description.

The figure shows the graph of the function f, defined by , f of x equals the absolute value of 2 x, end absolute value, plus 4 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. g of x = x minus 2

B. g of x = x + 3
C. g of x = 2 x minus 2
D. g of x = 2 x + 3
E. g of x = 3 x 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.

Begin Figure Description

This figure is the same as the figure accompanying the question 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.

End figure description.

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 = 3 x minus 2.

Note: This question also appears as a sample question for Strategy 3.

Strategy 7: Find a Pattern

Patterns are found throughout mathematics. Identifying a pattern is often the first step in understanding a complex mathematical situation. Pattern recognition yields insight that may point in the direction of a complete solution to the problem or simply help you generate a hypothesis, which requires further exploration using another strategy. In a problem where you suspect there is a pattern but don’t recognize it yet, working with particular instances can help you identify the pattern. Once a pattern is identified, it can be used to answer questions.

Sample Question for Strategy 7: Multiple-Choice – Select One or More Answer Choices Question.

Which of the following could be the units digit of 57 to the power n, where n is a positive integer?

Indicate all such digits.

1. 
   0
   
2. 
   1
   
3. 
   2
   
4. 
   3
   
5. 
   4
   
6. 
   5
   
7. 
   6
   
8. 
   7
   
9. 
   8
   
10. 
    9
    

Explanation

The units digit of 57 to the power n is the same as the units digit of 7 to the power n for all positive integers n. To see why this is true for n = 2, compute 57 to the power 2 by hand and observe how its units digit results from the units digit of 7 to the power 2. Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with n = 1 and proceeding consecutively, the units digits of 7, 7 to the power 2, 7 to the power 3, 7 to the power 4, and 7 to the power 5 are 7, 9, 3, 1, and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of 7 to the power n and therefore of 57 to the power n. The correct answer consists of the four choices B, D, H, and J, which are 1, 3, 7, and 9, respectively.

Note: This question also appears as a sample question for Strategy 12.