GMAT Math : Graphing

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #10 : Graphing A Line

Which of these equations is represented by a line that does not intersect the graph of the equation  ?

Possible Answers:

None of the other choices gives a correct answer.

Correct answer:

Explanation:

We can find out whether the graphs of  and  intersect by first solving for  in the first equation:

We then substitute in the second equation for :

Then we rewrite in standard form:

Since we are only trying to pdetermine whether at least one point of intersection exists, rather than actually find the point, all we need to do is to evaluate the discriminant; if it is nonnegative, at least one solution - and, consequently, one point of intersection - exists. In the general quadratic equation , this is , so here, the discriminant is

.

Therefore, the line of the equation  intersects the parabola of the equation .

We do the same for the other three lines:

 

Then we rewrite in standard form:

.

The line of  intersects the parabola.

 

The line of  intersects the parabola.

 

Since the discriminant is negative, the system has no real solution. This means that the line of  does not intersect the parabola of the equation , and it is the correct choice.

Example Question #41 : Graphing

A line with positive slope passes through the vertex and an -intercept of the parabola of the equation . What is the equation of the line?

Possible Answers:

Correct answer:

Explanation:

The vertex of the parabola of an equation of the form   has -coordinate . Here, we substitute , to obtain -coordinate

.

To find the -coordinate, substitute this for :

The vertex is .

To find the -intercepts of the parabola, substitute 0 for  in the equation:

Either

, in which case ,

or

, in which case .

The -intercepts are  and .

The line includes  and either  or , so we find the slope in each case using the slope formula.

If the line includes  and :

.

If the line includes  and :

We choose the first case, since the line has positive slope. The line through  and  has as its equation, using the point-slope form with  and point :

Example Question #42 : Graphing

The graph of the equation   shares its -intercept and one of its -intercepts with a line of positive slope. What is the equation of the line?

Possible Answers:

Correct answer:

Explanation:

The -intercept of the line coincides with that of the graph of the quadratic equation, which is a horizontal parabola; to find the -intercept of the parabola, substitute 0 for  in the quadratic equation:

The -intercept of the parabola, and of the line, is .

The -intercept of the line coincides with one of those of the parabola; to find the -intercepts of the parabola, substitute 0 for  in the equation:

Either

, in which case ,

or

, in which case .

The -intercepts of the parabola are  and , so the -intercept of the line is one of these. We will examine both possibilities.

If  and  be the - and -intercepts, respectively, of the line, then the slope of the line is . If the intercepts of the line are  and , the slope of the line is ; if the intercepts are  and , the slope is . We choose the latter, since we are looking for a line with positive slope; since its -intercept is , then we can substitute  in the slope-intercept form of the line, , to get the correct equation, .

 

Example Question #1021 : Problem Solving Questions

Give the -coordinate of the point of intersection of the lines of the equations:

Round your answer to the nearest whole number, if applicable.

Possible Answers:

The lines of the equations do not intersect.

Correct answer:

Explanation:

The point of intersection of the two lines has as its coordinates the values of  and  that make both of the given linear equations true. Therefore, we seek to find the solution of the system of equations:

We need only find , so multiply both sides of the two equations by 7 and 4, respectively. Then add:

              

,

making 9 the correct response.

Example Question #44 : Graphing

Line_1

Which of the following equations can be graphed with a line perpendicular to the green line in the above figure, and with the same -intercept?

Possible Answers:

Correct answer:

Explanation:

The - and -intercepts of the line are, respectively,  and .  If  and  are the - and -intercepts, respectively, of a line, the slope of the line is . This makes the slope of the green line 

Any line perpendicular to this line must have as its slope the opposite of the reciprocal of this, or . Since the desired line must also have -intercept , then the slope-intercept form of the line is

which can be rewritten in standard form:

Example Question #45 : Graphing

A triangle is formed by the -axis and the graphs of the equations 

and 

Give the area of the triangle.

Possible Answers:

Correct answer:

Explanation:

The vertices of the triangle are the point of intersection of the graphs of the lines of the two equations, and the -intercepts of those lines.

The -intercept of the line of the equation  can be found by setting  and solving for :

The -intercept of the line of the equation  can be found the same way:

The -intercepts are  and ; these are two of the vertices of the triangle, and since the segment connecting them is horizontal, this will be taken as the base. The length of the base is the difference of the -coordinates:

The intersection of the lines of the equations  and  can be found by solving the system of linear equations, as follows:

         

The point of intersection, and the third vertex of the triangle, is .

Since we are taking the horiztonal segment to be the base, the height will be the vertical distance to this third point - namely, the -coordinate . The area is half the product of the base and the height:

Example Question #1022 : Problem Solving Questions

Which response comes closest to the area of the triangle on the rectangular coordinate plane whose sides are along the axes and the line of the equation ?

Possible Answers:

Correct answer:

Explanation:

The intercepts of the line of the equation  can be found by substituting 0 for each variable, in turn:

-intercept:

The -intercept is the point 

 

-intercept:

The -intercept is the point .

This line and the axes together form a right triangle. The horizontal leg is the segment connecting the origin to , and its length is . The vertical leg is the segment connecting the origin to , and its length is . The area of a right triangle is half the product of these legs, which is 

.

Of the five responses, 45 comes closest to the correct area.

 

Example Question #1 : Graphing An Ordered Pair

Which quadrants can include the point   if nonzero numbers  and  have the same sign?

Possible Answers:

Either Quadrant I or Quadrant IV

Either Quadrant I or Quadrant II

Either Quadrant II or Quadrant III

Either Quadrant II or Quadrant IV

Either Quadrant I or Quadrant III

Correct answer:

Either Quadrant I or Quadrant IV

Explanation:

If  and  have the same sign, then , and the point has a positive -coordinate. The sign of  is the same as the common sign of  and . Therefore, only the -coordinate restricts the possibilities.

The points witn positive -coordinates are in Quadrant I and Quadrant IV.

Example Question #102 : Coordinate Geometry

Which quadrant or axis can include the point ?

Possible Answers:

Quadrant IV

The -axis

Quadrant III

Quadrant II

Quadrant I

Correct answer:

Quadrant I

Explanation:

The ordered pair  consists of a positive -coordinate and a positive -coordinate. The only quadrant that consists entirely of positive - and -coordinates is Quadrant I. 

Example Question #1031 : Gmat Quantitative Reasoning

Which quadrant or axis contains the point ?

Possible Answers:

The -axis

Quadrant III

Quadrant II

The -axis

Quadrant I

Correct answer:

The -axis

Explanation:

The ordered pair  consists of a positive -coordinate and a -coordinate of . The only part of the coordinate plane on which all points have a zero -coordinate is the -axis. 

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