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Unformatted text preview: KW”) 7:6? UJW‘ SK \2.“\ 1.9 Proportions and Power Functions
Power Functions y = Rx? . Note that polynomials are the sums of power functions
(which makes a power function also a polynomial). Determine if the following are power functions and if so, ﬁnd k and p 5
y = E;
y = 16x
_ i
y 2x
“a 2:
y 3
y — (3x3): x
y = 23: +1
Proportionality
Ex (13) The energy, E, expended by a swimming dolphin is proportional to the cube of
the speed, v, of the dolphin. ,1)
a = v V
If a dolphin uses 500 caloriesfhour swimming at 20 mph, how many calories per hour
W111 it use sw1mm1ng 30 mph? so.) 940‘) ‘ , if
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#\q’] E f D C} \l\ a ‘5 H95? 7 5 (gt/Hoot?— A directly proportional function will look like y=k*(?), while an inversely
proportional function will look like: y=k*1;’(?) (note that the k is never ﬂipped). Ex (19) Biologists estimate that the number of animal species of a certain body length is
inversely proportional to the square of the body length.
A) Write a formula for the number of animal Species, N, of a certain body length as a function of the length, L.
N = WA B) If there are 100 species that are 20 feet long, according to the formula, how many
species are 1 inch long? . } ‘z
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6303,5300 Polynomials
Most of you already know what a polynomial is, but we want to clarify some Hi“ we
deﬁnitions. Let’s take as an example: f(x) = 4x2 +2x —3x“ +4 : ﬂy“ 4 ti? +2»; “4/7 Law I 1) the degree of a polynomial is the highest power that you See in the problem
2) the leading coefﬁcient is the number in front of the x with the highest power If we look at the graph up close, we see several things going on. The power ofa polynomial is at least one more than the number of local extrema “bumps or turning
points ” V f\ 14 o ' a . — A /\ 3
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_§\ﬁ When we look at it zoomed out, it just looks like the power ﬁmction consisting of the leading coefﬁcient and x to the degree. Note that if the degree of the polynomial is
Odd: Even: The leading coefﬁcient also has an effect. The macro effect is only determined by the
sign: What do an odd and even graphs look like if the leading coefﬁcient is negative? So in the short run, a large leading coefﬁcient will make the graph grow fast, but in the
long run, the growth is determined by: m at =33: “‘“=="':‘1€«
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