General Problem-Solving Steps


Strategy 2: Translate from Words to a Figure or Diagram



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Strategy 2: Translate from Words to a Figure or Diagram


To solve a problem in which a figure is described but not shown, draw your own figure. Draw the figure as accurately as possible, labeling as many parts as possible, including any unknowns.

Drawing figures can help in geometry problems as well as in other types of problems. For example, in probability and counting problems, drawing a diagram can sometimes make it easier to analyze the relevant data and to notice relationships and dependencies.


Sample Question for Strategy 2: Multiple-Choice – Select One Answer Choice Question.


Which of the following numbers is farthest from the number 1 on the number line?

A. negative 10

B. negative 5

C. 0


D. 5

E. 10

Explanation


Circling each of the answer choices in a sketch of the following number line shows that of the given numbers, negative 10 is the greatest distance from 1.


Begin figure description.

The figure is a number line with 23 equally spaced tick marks labeled with the integers from negative 11 through positive 11. Going from left to right, the 5 evenly spaced integers negative 10, negative 5, 0, 5, and 10 are circled. The integer 1 is 1 tick mark to the right of 0.
End figure description.

Another way to answer the question is to remember that the distance between two numbers on the number line is equal to the absolute value of the difference of the two numbers. For example, the distance between negative 10 and 1 is the absolute value of negative 10 minus 1, which equals 11 and the distance between 10 and 1 is the absolute value of 10 minus 1, which equals the absolute value of 9, which equals 9. The correct answer is Choice A, . negative 10.

Strategy 3: Translate from an Algebraic to a Graphical Representation


Many algebra problems can be represented graphically in a coordinate system, whether the system is a number line if the problem involves one variable, or a coordinate plane if the problem involves two variables. Such graphs can clarify relationships that may be less obvious in algebraic presentations.

Sample Question for Strategy 3: Multiple-Choice – Select One Answer Choice Question.


This question is based on the following figure.


Begin figure description.

The figure shows the graph in the xy-plane of the function f of x = 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 = the absolute value of 2 x, end absolute value, + 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 ?

  1. g of x = x minus 2

  2. g of x = x + 3

  3. g of x = 2 x minus 2

  4. g of x = 2 x + 3

  5. 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 6.



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