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Example Questions
Example Question #51 : Coordinate Geometry
Give the -coordinate of the
-intercept of the graph of the function
The -intercept(s) of the graph of
are the point(s) at which it intersects the
-axis. The
-coordinate of each is 0,; their
-coordinate(s) are those value(s) of
for which
, so set up, and solve for
, the equation:
Add 8 to both sides:
Divide both sides by 2:
Take the common logarithm of both sides to eliminate the base:
Example Question #51 : Graphing
Give the domain of the function .
The set of all real numbers
The set of all real numbers
Let . This function is defined for any real number
, so the domain of
is the set of all real numbers. In terms of
,
Since is defined for all real
, so is
; it follows that
is as well. The correct domain is the set of all real numbers.
Example Question #52 : Graphing
Give the range of the function .
The set of all real numbers
Since a positive number raised to any power is equal to a positive number,
Applying the properties of inequality, we see that
,
and the range of is the set
.
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.
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