Geometry › 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 -intercept
IV) The locations of the -intercepts, if there are any
V) The equation of the line of symmetry
I and III only
I and V only
I, II, and V only
I, III, and IV only
III and IV 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.
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 .
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 .
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.
Give the -coordinate of the
-intercept of the graph of the function
The graph of has no
-intercept.
The -intercept of the graph of
is the point at which it intersects the
-axis. Its
-coordinate is 0; its
-coordinate is
, which can be found by substituting 0 for
in the definition:
,
the correct choice.
Find the -intercept and range for the function:
Find the equation based on the graph shown below:
When you look at the graph, you will see the x-intercepts are
and the y-intercept is
.
These numbers are the solutions to the equation.
You can work backwards and see what the actual equation will come out as,
.
This would distribute to
and then simplify to
.
This also would show a y-intercept of .
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.
Determine the domain and range for the graph of the below function:
When finding the domain and range of a quadratic function, we must first find the vertex.