Transcript 2 1 Quadratic Functions

Objectives:
1. To find the vertex,
axis of symmetry,
and zeros of
parabolas
2. To convert quadratic
functions to
“standard” form
•
•
•
•
•
Assignment:
P. 134: 15-27 odd
P. 135: 43, 45, 49
P. 136: 78, 79
Homework
Supplement
Read: P. 139-147
Quadratic Function
Parabola
Vertex
Axis of Symmetry
Zeros
Roots
Perfect Square
Trinomials
Completing the Square
Put the quadratic functions in order from widest
to narrowest graph.
3
1
y  3x 2 , y   x 2 , y  x 2 , y  2 x 2
2
2
A quadratic function can
be written in the form
y  ax 2  bx  c
where a, b, and c are real
numbers and a ≠ 0.
• Quadratic functions
are polynomial
functions of degree 2.
A quadratic function can
be written in the form
y  ax 2  bx  c
where a, b, and c are real
numbers and a ≠ 0.
• Vertex: The turning
point of the parabola;
marks the highest or
lowest point
Vertex
where a, b, and c are real
numbers and a ≠ 0.
• Axis of Symmetry:
Line through the
vertex that divides the
parabola in “half”
Axis of Symmetry
A quadratic function can
be written in the form
y  ax 2  bx  c
For the graph of y = ax2 + bx + c :
Parameter
Behavior on the Graph
1.
a
2.
1.
c
Determines width of parabola
• As |a| increases, the parabola gets narrower
• As |a| decreases, the parabola flattens out/gets
wider
Determines upward or downward shape
• If a > 0,  (beard U)
• If a < 0,  (mustache H)
Determines the y-intercept of the parabola
• As c increases, the parabola moves up
• As c decreases, the parabola moves down
Use SRT transformations
to graph the quadratic
function below.
y  2  x  3  4
2
Use SRT transformations
to graph the quadratic
function below.
y  2  x  3  4
2
Where is the vertex?
Where is the axis of
symmetry?
You will be
able to convert
quadratic functions
to “standard” form
The quadratic function below is in standard
form (formerly known as Vertex Form):
f ( x)  a( x  h)2  k
• Vertex: (h, k)
• Axis of symmetry: x = h
How do we get our function into this form?
What we want to do is
to turn an expression
like x2 + 6x into a
perfect square
trinomial.
To understand the
process of
completing the
square, we will
consider a geometric
interpretation.
Geometric representation of x2:
x
x
x2
x:
1
x x
Unit Square:
1
1 1
So you could represent x2 + 6x geometrically
with the tiles below, but it doesn’t look much
like a square.
x2
x
x
x
x
x
x
We make this expression into a square by
rearranging these pieces and adding some
unit squares.
x+3
x x x
2
2
x x x
x+3
x2
To create this square,
put half of your x’s on
one side of the
= (x + 3)
square and the other
= x + 6x + 9
half of your x’s on the
other side of the
square. Now fill in
the missing bits with
unit squares.
1 1 1
1 1 1
1 1 1
x+3
x x x
2
x x x
2
1 1 1
1 1 1
1 1 1
half
x+3
x2
In general then, to
complete the square,
you take half of the
= (x + 3)
middle term and then
= x + 6x + 9
square it. The
number you get
3
becomes the
constant term.
x+3
x x x
2
x x x
2
1 1 1
1 1 1
1 1 1
half
x+3
x2
Notice also that the
number you get from
taking half of the
= (x + 3)
middle term becomes
= x + 6x + 9
the constant of the
binomial that gets
3
squared.
Find the value of c that makes x2 – 26x + c a
perfect square trinomial. The write the
expression as the square of a binomial.
Find the value of c that makes x2 + 7x + c a
perfect square trinomial. The write the
expression as the square of a binomial.
Find the value of c that makes the expression a
perfect square trinomial. Then write the
expression as the square of a binomial.
1. x2 + 14x + c
2. x2 – 22x + c
3. x2 – 9x + c
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y  x  6 x  12
2
There can be 2, 1,
or 0 zeros for a
quadratic
function,
depending upon
where it is in
the coordinate
plane.
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y  x  10 x  5
2
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y  x  9x  2
2
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y   x  3x  1
2
Find the value of c that makes 3x2 – 36x + c a
perfect square trinomial. The write the
expression as the square of a binomial.
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y  2 x  4 x  14
2
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
y  3x  x  1
2
Put the function shown
into standard form.
Then find the vertex,
the equation of the
axis of symmetry, the
min/max value, and
the zeros.
1 2
y   x  3x  6
2
Write the equation of the parabola that has the
vertex (4, -1) and whose graph passes through
the point (2, 3).
Write the equation of the parabola that has the
vertex (2, 3) and whose graph passes through
the point (0, 2).
Objectives:
1. To find the vertex,
axis of symmetry,
and zeros of
parabolas
2. To convert quadratic
functions to
“standard” form
•
•
•
•
•
Assignment:
P. 134: 15-27 odd
P. 135: 43, 45, 49
P. 136: 78, 79
Homework
Supplement
Read: P. 139-147