September 18, 2014
Chapter 2 Polynomial and Rational Functions
Section 2.1 Quadratic Functions
Objective: Learn how to sketch and analyze graphs of quadratic functions.
Important Vocabulary (that you should already know!)
Constant function
A polynomial function with degree 0. f(x) = a, a = 0
2
Lin
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Word Bank
Vertex
Polynomial function
f(x)=a x +a x +...+a x+a
n
ea
un
rf
n ct
ion
Constant function
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Quadratic function
A polynomial function with degree 2.
f(x) = ax + bx + c, a = 0
Axis of symmetry
Qu
ad
Linear function
A polynomial function with degree 1.
f(x) = mx + b, m = 0
n
n-1
n-1
1
0
(a is a real #, n is a nonnegative integer)
e...
s
e
th
y=f(x+c)
er
b
m
e
y=f(x)+c
Rem
Note: n=0 --> constant
n=1 --> linear
n=2 --> quadratic
1
y=-f(x)
y=f(-x)
y=af(x)
Horizontal shift
Vertical shift
Refection in x-axis
Reflection in y-axis
Vertical Stretch/shrink
Describe the transformations:
1. f(x)= .25x2 2. g(x)= 3x2
3. k(x)= -x2+1 4. h(x)= (x+2)2-3
a > 1 stretch, 0< a <1 shrink
September 18, 2014
Standard Form (Vertex Form)
f(x) = a(x-h)2 + k, a = 0
Vertical axis:
x=h
Vertex:
(h, k)
If a > 0, parabola opens up (vertex = min. )
If a < 0, parabola opens down (vertex = max.)
Example:
1. Find the vertex of the following parabola.
f(x)= -2x2 - 4x + 1
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H
2. Graph the following quadratic function.
f(x)= x2 - 4x -2
To write a quadratic function in standard
form... Complete the square!
Ex1: f(x) = -2x2 - 4x + 1
Ex2: f(x) = x2- 4x-2
September 18, 2014
Ex3: Find the standard form of the equation of the parabola
that has vertex (1, -2) and passes through (3, 6).
a!
FinHdint
Ex4: Describe the graph and identify any x-intercepts.
f(x) = -x2+6x-8
To find x-intercepts (if exist) use...
factoring, quadratic formula, or graph.
Try completing the square on the general form
f(x)=ax2+ bx+c
Note: You can find the x-coordinate of the vertex by
finding -b and substituting back in to find y.
2a
September 18, 2014
Graph y=(x-4)2+5.
What is the minimum value of y?
Graph
Ex5: The daily cost of manufacturing a particular product is
given by C (x) = 1200 - 7x + 0.1x2 where x is the number of
units produced each day. Determine how many units should be
produced daily to minimize cost.
Algebraic Solution
We need to find h.
Graphical solution
Answer
Producing 35 units per day will minimize cost.