Download IB 2.1 Quadratic Functions

3/21/2017
Chapter 2:
Polynomial and
Rational
Functions
Power Standard #7
2.1 Quadratic
Functions
Lets glance at the finals.
Learning Objective: In this lesson you learned how to
sketch and analyze graphs of quadratic functions.
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The Graph of a Quadratic
Function
•
Definition of a Polynomial function:
•
Let n be a nonnegative integer and let , – , . . . , , , be
real numbers with 0. A polynomial function of x with
degree n is . . . the function
– – . . . .
What we really need to get out of this is that the order
of polynomial functions lead with the highest exponent
of the first letter in the alphabet. ( The Graph of a Quadratic
Function
•
Quadratic functions have a degree polynomial of 2, or a Second
Degree polynomial.
•
The graph of a quadratic is a U shape, which is called a
parabola.
•
, 0.
•
A couple of characteristics:
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If the leading coefficient is +
the function opens upward with a
minimum point. If the leading
coefficient is – then the function
opens downward and has a maximum
point.
•
Domain: ∞, ∞
•
Range: , ∞ , ∞, , depending on which
direction it opens to.
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The Graph of a Quadratic
Function
•
Standard Form of a Quadratic:
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f() = – ℎ
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Where: The graph of f is a parabola with a vertical
axis of symmetry, x = h. What does that mean?
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The axis of symmetry is the ℎ = −
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The vertex is at (h, k). If a > 0, the parabola opens
upward, and if a < 0, the parabola opens downward.
+ , ≠ 0
The process to find this form is to completing the
square.
So we need to solve + + = 0
Completing the Square review
ࢌ ࢞ ൌ ૛࢞૛ ൅ ૡ࢞ ൅ ૞
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1: make f(x)=0 and
subtract the constant.
•
2: Take of GCF of the
numbers
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3: Take b, divide it by
2, then square it.
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4: make the perfect
square with the value. Add
·
to the left side.
•
1: 0 2 8 5
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2:5 4
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3:
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4: 5 2 4 32 2 •
5: 3 2 2 ! " " #
•
5: Add/subtract the
left side back and put f(x)
back.
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Finding x & y intercepts
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Remember that x-intercepts exist on the x axis, so this is where
$ 0. We plug 0 in for y and solve.
•
You may need to factor, use the square root method, or complete
the square to find your x-intercepts. There may not be any real
solutions.
•
To find the y-intercept, simply set 0 and solve for y. If you have
a constant in an equation, it is usually the y-intercept.
The Graph of a Quadratic
Function
•
Example find the vertex, axis of symmetry & x & y intercepts of the
parabola. Graph these and a few points.
•
6 8
CTS:
Vertex:
AOS:
y-intercept:
x-intercept:
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Your turn: OYO
•
Sketch the graph of ଶ 2 8 and identify the
vertex, axis, and y & x-intercepts of the parabola.
Graph a few points.
Vertex:
AOS:
y-intercept:
x-intercept:
Your turn: Answer!
• Sketch
the graph of 2 8 and
identify the vertex, axis, and x-intercepts of the
parabola.
• (-
1, - 9);
• AOS:
• (•
x = - 1;
4, 0) and (2, 0)
0, 8
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OYO Again without graphing
ଶ 4 2
• Find:
• AOS:
• Vertex:
• y-int:
• x-int:
The Graph of a Quadratic
Function Example of finding a.
•
Find the standard form of the equation of the parabola that
has vertex at (1, -2) and passes through the point (3, 6).
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From the vertex we have this much of the equation: %() =
– 1 – 2. To find a we substitute the point (3, 6) and solve
for a.
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The Graph of a Quadratic
Function
•
Graph $ – 4 5. what is the minimum value of y?
•
The minimum occurs at
െ
௕
as well as the maximum. This
ଶ௔
is also at the vertex’s x value in a quadratic. So…
•
Evaluating the function at for is another way to find the
axis of symmetry, the x value in the vertex, and the where the
minimum or maximum value is of a quadratic.
The Graph of a Quadratic
Function: a rough one.
•
Find the minimum value of the function without a calculator.
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%() = 3 − 11 + 16.
•
At what value of x does this minimum occur? What is the
minimum point! Completing the square my be too rough.
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Max Height of a Projectile
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The path of a softball is given by the function
% = −0.007 + + 4,Where %() is the height of the baseball (in
feet) and x is the horizontal distance from home plate (in feet).
What is the maximum height reached by the ball?
•
We’ll need to use the equation for the axis of symmetry.
−
,%
−
Application
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The daily cost of manufacturing a particular product is
•
given by
•
'() = 1200– 7 + 0.1 •
Where x is the number of units produced each day. Find how
many units should be produced to minimize cost.
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' = 0.1 − 7 + 1200
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−1200 = 0.1( − 70)
•
−
= −35
= 1225
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−1200 + 122.5 = 0.1 − 35
•
' = 0.1 − 35
+ 1077.5
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Application
•
The daily cost of manufacturing a particular product is
•
given by
•
' 1200– 7 0.1 •
Where x is the number of units produced each day. Find how
many units should be produced to minimize cost.
•
= 0.1 35
1077.5
Producing 35 units per day will
minimize cost.
Homework
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P.96
•
#12, 21-29odd, 31, 35, 39, 41, 51, 65, 70
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