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**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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