All GMAT Math Resources
Example Questions
Example Question #10 : Graphing An Exponential Function
Evaluate
.
The system has no solution.
Rewrite the system as
and substitute
and for and , respectively, to form the system
Add both sides:
.
Now backsolve:
Now substitute back:
and
Example Question #1 : Graphing A Quadratic Function
What are the possible values of
if the parabola of the quadratic function is concave upward and does not intersect the -axis?
The parabola cannot exist for any value of
.
The parabola cannot exist for any value of
.If the graph of
is concave upward, then .If the graph does not intersect the
-axis, then has no real solution, and the discriminant is negative:
For the parabola to have both characteristics, it must be true that
and , but these two events are mutually exclusive. Therefore, the parabola cannot exist.Example Question #751 : Geometry
Which of the following equations has as its graph a vertical parabola with line of symmetry
?
The graph of
has as its line of symmetry the vertical line of the equation
Since
in each choice, we want to find such that
so the correct choice is
.Example Question #1 : Graphing A Quadratic Function
Which of the following equations has as its graph a concave-right horizontal parabola?
None of the other responses gives a correct answer.
A horizontal parabola has as its equation, in standard form,
,
with
real, nonzero.Its orientation depends on the sign of
. In the equation of a concave-right parabola, is positive, so the correct choice is .Example Question #1 : Graphing A Quadratic Function
The graphs of the functions
and have the same line of symmetry.If we define
, which of the following is a possible definition of ?
None of the other responses gives a correct answer.
The graph of a function of the form
- a quadratic function - is a vertical parabola with line of symmetry .The graph of the function
therefore has line of symmetry, or
We examine all four definitions of
to find one with this line of symmetry.
:
, or
:
, or
, or
, or
Since the graph of the function
has the same line of symmetry as that of the function , that is the correct choice.Example Question #1 : How To Graph A Quadratic Function
Give the
-coordinate of a point at which the graphs of the equations
and
intersect.
We can set the two quadratic expressions equal to each other and solve for
.and , so
The
-coordinates of the points of intersection are 2 and 6. To find the -coordinates, substitute in either equation:
One point of intersection is
.
The other point of intersection is
.
1 is not among the choices, but 41 is, so this is the correct response.
Example Question #51 : Coordinate Geometry
Give the set of intercepts of the graph of the function
.
The
-intercepts, if any exist, can be found by setting :
The only
-intercept is .
The
-intercept can be found by substituting 0 for :
The
-intercept is .
The correct set of intercepts is
.Example Question #7 : Graphing A Quadratic Function
Give the
-coordinate of a point of intersection of the graphs of the functions
and
.
The system of equations can be rewritten as
.
We can set the two expressions in
equal to each other and solve:
We can substitute back into the equation
, and see that either or . The latter value is the correct choice.Example Question #1 : How To Graph A Quadratic Function
has as its graph a vertical parabola on the coordinate plane. You are given that and , but you are not given .
Which of the following can you determine without knowing the value of
?I) Whether the graph is concave upward or concave downward
II) The location of the vertex
III) The location of the
-interceptIV) The locations of the
-intercepts, if there are anyV) The equation of the line of symmetry
III and IV only
I and V only
I and III only
I, II, and V only
I, III, and IV only
I and III only
I) The orientation of the parabola is determined solely by the sign of
. Since , the parabola can be determined to be concave downward.II and V) The
-coordinate of the vertex is ; since you are not given , you cannot find this. Also, since the line of symmetry has equation , for the same reason, you cannot find this either.III) The
-intercept is the point at which ; by substitution, it can be found to be at . known to be equal to 9, so the -intercept can be determined to be .IV) The
-intercept(s), if any, are the point(s) at which . This is solvable using the quadratic formula
Since all three of
and must be known for this to be evaluated, and only is known, the -intercept(s) cannot be identified.The correct response is I and III only.
Example Question #2 : How To Graph A Quadratic Function
Which of the following equations can be graphed with a vertical parabola with exactly one
-intercept?
The graph of
has exactly one -intercept if and only if
has exactly one solution - or equivalently, if and only if
Since in all three equations,
, we find the value of that makes this statement true by substituting and solving:
The correct choice is
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