Unformatted text preview: Lesson 25
Quadratic Function:
 a function is quadratic if it is in any of the following forms:
o ()
, where
(
)
o ()
, where
(
)(
), where
o ()
 the degree of a quadratic function is _______
 the graph of a quadratic function a parabola
Vertex of a parabola:
 the turning point, the point where the graph of a quadratic function
changes from increasing to decreasing, or vice versa
 the coordinate of the vertex is either the maximum or minimum
function value
 the coordinate of the vertex is the input where the maximum or
minimum function value occurs
Example 1: Graph the following quadratic functions. List the vertex and
any symmetry.
a. ( )
b. ( ) b. ( ) The graph of a quadratic function is a parabola that opens up or down
(why can’t it open to the left or right?).
If the leading coefficient
, the parabola opens up; if the leading
coefficient
, the parabola opens down (why can’t the leading
coefficient
?).
The graph of a quadratic function will always be symmetric about a
vertical line called the line of symmetry.
Example 2: Graph the following functions by transforming the graph of
the function ( )
. Find the vertex of each and list any symmetry.
)
(
)
a. ( ) (
b. ( ) c. ( ) Can you determine the vertex of each graph without graphing? Can you determine the line of symmetry for each graph without graphing? Standard Equation of a Parabola with Vertical Axis:
(
)
)
 the vertex is (
 when
, the parabola opens _______
 when
, the parabola opens _______
 symmetric about the line
, where is the coordinate of the
vertex
When a quadratic function, such as
()
is written in the form of the standard equation of a parabola with a vertical
axis
( ) the vertex is ( ), the line of symmetry is
sketched by transforming the basic parabola
units to the right and 2 units up). , and the graph can be
(shift the graph 3 How can a quadratic function of the form ( )
(
)
converted to the form ( )
?
How can ( ) be written in the form be
( ) ? Example 3: Given the following functions, find the standard equation of
the parabola, then graph and find the following:
a. ( ) ()
() when b. ( ) ()
() when c. ( ) ()
() d. ( ) ()
() when ( when )( ) Example 4: Find the standard equations of the parabolas shown:
a.
b. c. d. Looking back at the graphs in previous two examples, do you notice a
relationship between the coordinate of the vertex and the intercepts? Example 5: Find the standard equation of a parabola with a vertical axis
that satisfies the given conditions:
)
a. Vertex ( ), passing through ( b. Vertex ( c. intercepts ), intercept and , minimum value of ...
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 Spring '08
 Algebra, Trigonometry, Quadratic equation, Conic section

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