**Quadratic Functions**

**Sol 1** The x-coordinate is given by
, and
then the y-coordinate is given by
.

**Sol 2** The x-coordinate of the vertex is given by
; so (since ) the maximum value is
given by
.

**Sol 3** Since 5 and 1 are the x-intercepts of the graph,
for some number . Since the graph of is symmetric
around the vertical line passing through its vertex, the x-coordinate of the
vertex is 3 (the average of 5 and 1). Therefore the minimum value of is
, so and
or
.

**Sol 4** Since the vertex is at , the parabola has an equation of
the form
or . Since the parabola passes through
the point , substituting and gives , so
and
, so
.

**Sol 5**
Substituting in and replacing by gives the
quadratic function
. Then the t-coordinate of the vertex of
the graph of is given by
, so the minimum
value of is given by
Therefore the minimum value of is also 9, and this value is attained when
so .

**Sol 6** Since the roles of and are interchanged, the y-coordinate
of the vertex is given by
and the
x-coordinate is given by
.

**Sol 7** Since is the minimum value of , the graph of has
an equation of the form
or with .
Since , substituting and gives and
therefore . Therefore , so
or
.

**Sol 8** Since the line is non-vertical and passes through the point ,
it has an equation of the form or . Since the line
intersects the parabola exactly once, the equation or
has exactly one solution. Therefore the discriminant
is equal to 0, so
,so the line has
equation or .

**Sol 9** (If the horizontal line intersects the graph of only once,
then will correspond to the maximum value or minimum value of .) Setting
gives
, so and therefore
. This equation will have exactly one solution if the
discriminant is zero, so
.
Therefore has a maximum value of 3, since
has no solution.

Return to the Problems for this Topic.

Return to Precalculus Home Page.