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Unit 4: Building Polynomial and Rational Functions/ Equations
“UNPACKED STANDARDS”
N.Q.1 Interpret units in the context of the problem
N.Q.1 When solving a multi-step problem, use units to evaluate the
appropriateness of the solution.
N.Q.1 Choose the appropriate units for a specific formula and interpret the
meaning of the unit in that context.
N.Q.1 Choose and interpret both the scale and the origin in graphs and data
displays
N.Q.2 Determine and interpret appropriate quantities when useing descriptive
modeling.
N.Q.3 Determine the accuracy of values based on their limitations in the context
of the situation.
F.IF.2 When a relation is determined to be a function, use f(x) notation.
F.IF.2 Evaluate functions for inputs in their domain.
F.IF.2 Interpret statements that use function notation in terms of the context in
which they are used.
F.IF.4 Given a function, identify key features in graphs and tables including:
intercepts; intervals where the function is increasing, decreasing, positive, or
negative; relative maximums and minimums; symmetries; end behavior; and
periodicity.
F.IF.4 Given the key features of a function, sketch the graph.
F.IF.7c Polynomial functions, identifying zeros when factorable, and showing end
behavior.
Ex: Consider the function f (x) = x3 – 4x2 – 3x + 1
(a) Find the coordinates of the maximum and minimum points of the
function.
(b) Find the values of f (x) for a and b in the table below:
x
–3 –2 –1
0
1
2
3
4
5
f (x) –36 a
b
16
12
4
0
6
28
(c) Using a scale of 1 cm for each unit on the x-axis and 1 cm for
each 5 units on the y-axis, draw the graph of f (x) for –3 ≤ x ≤ 5.
Label clearly.
Ex : (a) Sketch the curve of the function f (x) = x3 − 2x2 + x − 3 for values
of x from −2 to 4, giving the intercepts with both axes.
(b) On the same diagram, sketch the line y = 7 − 2x and find the
coordinates of the point of intersection of the line with the curve.
Ex: The velocity, vms–1, of a kite, after t seconds, is given by
4t2 + 4t,
0 < t< 4.
What is the velocity of the kite after
(i) one second?
(ii) half a second?
(b) Calculate the values of a and b in the table below.
t
0
0.5
1
1.5
2
2.5
3
v
a
b
0
0 0.625
v = t3 –
(a)
(c)
3.5
7.88
4
16
On graph paper, draw the graph of the function v = t3 – 4t2 + 4t,
0 < t< 4.
Use a scale of 2 cm to represent 1 second on the horizontal axis
and 2 cm to represent 2ms–1 on the vertical axis.
(d) Describe the motion of the kite at different times during the first 4
seconds.
Write down the intervals corresponding to changes in motion.
Ex: a) Sketch the graphs of the functions described by f(x)=x2 and g(x)=x4 on
the same axes, being careful to label any points of intersection. Also, find
and label (1 2 ,f(1 2 )) and (1 2 ,g(1 2 )) .
b) Sketch the graphs of the functions described by f(x)=x3 and g(x)=x5 on
the same axes, being careful to label any points of intersection. Also, find
and label (1.2,f(1 2 )) and (1 2,g(1 2 )) .
c) Sketch the graphs of the functions described by f(x)=x2 and g(x)=x3on the
same axes, being careful to label any points of intersection. Also, find and
label (1 2 ,f(1 2 )) and (1 2 ,g(1 2 )) .
F.IF.5 Given the graph of a function, determine the practical domain of the
function as it relates to the numerical relationship it describes.
Ex: The Profit, P(x), a company makes from selling gadgets, x, can be modeled
by the following equation: P(x) = -.01x4 + 2.3x3 + 2x2 – 500. Find the
practical domain of P(x) assuming the company will stop selling gadgets
when it stops making a profit.
N.CN.9 Understand The Fundamental Theorem of Algebra, which says that the
number of complex solutions to a polynomial equation is the same as the degree
of the polynomial. Show that this is true for a quadratic polynomial.
Ex:
a) How many zeros does f(x) =  2x 2  3x  8 have?
b) Find all the zeros and explain, orally or in written format, your answer in
terms of the Fundamental Theorem of Algebra.
Ex: How many complex zeros does the following polynomial have? How do you
know?
p( x )  ( x 2  3) ( x 2  2) ( x  3)2x  1
Ex: How many complex zeros does the following polynomial have? How do you
know?
A.APR.2 Understand and apply the Remainder Theorem.
The Remainder theorem says that if a polynomial p(x) is divided by x – a,
then the remainder is the constant p(a). That is, p(x)=q(x)(x - a)+ p(a). So
if p(a) = 0 then p(x) = q(x)(x-a).
Ex: Let p(x)= x 5 -3x 4 +8x 2 - 9x+30 .
a) Evaluate p(-2).
b) What does your answer tell you about the factors of p(x)?
Ex: Given a polynomial p(x) and a number a, use the Remainder Theorem to
determine if (x – a) is a factor of p(x).
Ex: Given a polynomial p(x) and a factor of p(x), (x-a), use the Remainder
Theorem to determine p(a) without substitution.
A.APR.2 Understand how this standard relates to A.SSE.3a.
A.APR.2 Understand that a is a root of a polynomial function if and only if x-a is a
factor of the function.
Ex: Consider the polynomial function P(x)=x4 −3x3 +ax2 −6x+14, where a is an
unknown real number. If (x−2) is a factor of this polynomial, what is
the value of a ?
A.APR.3 Find the zeros of a polynomial when the polynomial is factored.
A.APR.3 Use the zeros of a function to sketch a graph of the function
Ex: Factor the expression x 3 + 4x 2 - 59x -126 and explain how your answer can
be used to solve the equation x 3 + 4x 2 - 59x -126 = 0 . Explain why the solutions to
this equation are the same as the x-intercepts of the graph of the function
f (x) = x 3 + 4x 2 - 59x -126 .
Ex: f(x) has double root at -3 and two imaginary solutions. f(2) is 0. The y
intercept of f(x) is 8.
a) Give a rough sketch of f(x).
b) Give a partial factorization of f(x).
A.CED.1 Create linear, quadratic, rational and exponential equations and
inequalities in one variable and use them in a contextual situation to solve
problems.
Ex: Mike likes to canoe. He can paddle 150 feet per minute. He is planning a
river trip that will take him to a destination about 30,000 feet upstream
(that is, against the current). The speed of the current will work against the
speed that he can paddle.
1. Let s be the speed of the current in feet per minute. Write an expression
for r(s) , the speed at which Mike is moving relative to the river bank, in
terms of s .
2. Mike wants to know how long it will take him to travel the 30,000 feet
upstream. Write an expression for T(s) , the time in minutes it will take, in
terms of s .
3. What is the vertical intercept of T ? What does this point represent in
terms of Mike’s canoe trip?
4. At what value of s does the graph have a vertical asymptote? Explain why
this makes sense in the situation.
5. For what values of s does T(s) make sense in the context of the problem?
Ex: Sally can paint a classroom in 7 hours. John can paint a classroom in 5
hours. How long will it take them to paint a classroom if they work together?
A.CED.2 Create equations in two or more variables to represent relationships
between quantities.
A.CED.2 Graph equations in two variables on a coordinate plane and label the
axes and scales.
A.REI.2 Solve simple rational and radical equations in one variable and provide
examples of how extraneous solutions arise.
Ex: Solve the following equation, noting and explaining any extraneous solutions
10
4
5
 
x2 x x2
A.REI.1 Assuming an equation has a solution, construct a convincing argument
that justifies each step in the solution process. Justifications may include the
associative, commutative, and division properties, combining like terms,
multiplication by 1, etc.
Ex: Solve:
. Justify each step.