Transcript Section 2-1

Section 2-1
Linear and Quadratic
Functions
Section 2-1
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polynomial functions
linear functions
rate of change
linear correlation
quadratic functions
vertical motion problems
Polynomial Functions
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the only operations on the variable
are mult., addition, and subtraction
examples of poly.’s
f ( x)  3x  1
f ( x)  6 x  3x  8
2
f ( x)  3x  x  8 x  19
5
4
2
Polynomial Functions
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functions which are not polynomials
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variable
variable
variable
variable
under a square root
in an exponent
in a denominator
with a negative exponent
f ( x)  x  4
f ( x)  4 x
5
f ( x)  3
x2
4
f ( x) 
x 1
Linear Functions
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in the form f ( x)  mx  b
m is the slope and b is the
y-intercept
linear functions have degree = 1
problems might involve the use of
the slope formula, slope-intercept
form, and point-slope form
Rate of Change
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the average rate of change between
two values (x = a and x = b) is given
by the following equation:
f (b)  f (a)
ba
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for a linear function, the average
rate of change is constant (the slope)
and the initial value is the y-intercept
Linear Correlation
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when points of a scatter plot are
clustered around a line we say there
is a “linear correlation” between the
quantities represented by the data
if the regression line has positive
slope then there is a “positive
correlation” and r > 0 (same for
negative and r < 0)
if r is near 1 then there is strong
correlation and near 0 is weak or no
correlation
Quadratic Functions
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the graph of a quadratic function is
called a parabola
graphs can be sketched using the
normal transformations if the
equation is written in vertex form
f ( x )  a ( x  h)  k
vertex (h, k )
a  0 concave up
a  0 concave down
2
Quadratic Functions
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if the equation is in standard form
then finding the vertex takes a little
work:
2
y  ax  bx  c
b
h
2a
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k  c  ah
2
to convert from standard form to
vertex form you must complete the
square
Vertical Motion
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a special quadratic equation is used to
track the height of an object (that is
moving vertically) versus time
1 2
s(t )   gt  v0t  s0
2
s(t )  height at time t
v0  intial velocity s0  intial height
g = 32 ft/sec or 9.8 m/sec
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g is the force due to gravity and which
value you use depends on the units
Vertical Motion
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there is a linear equation which can
track the vertical velocity of the
object at a given time
v(t )   gt  v0
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you can use these functions to find
out the maximum height the object
can reach, when it hits the ground,
or its velocity when it hits the ground
(or other things as well)