PART I lesson, inquiry mode
1) consider the functions:
f(x) = x4 - 4x3 + 16x -16 g(x) = x4 -4x3 + 16x h(x) = x4+ x3 - 8x2 - 12x
These functions are ______ degree polynomial, the leading term is ____, the leading coefficient
is _____, the maximum number of x-intercepts is ______, the long-run behavior is:
Enter the functions in your TI as Y1, Y2, Y3. Set the window as : -10<x<10 and -200<y<200
trace the graphs. The 3 graphs look the same. right ?
What happens if you change a smaller scale. Use Zoom zbox to frame the area that includes the x-intercepts. (that is about -10<x<10 and -40<y<40). Do they still look alike ?
The 3 behaviors differ now. The intercepts are not the same so the bumps don't look alike. We say the long-run is the same because the leading term is the same but not the short-run. To study the short run, we need to study the area around the x-intercepts.
2) y = f(x) = x3 - x2 - 6x
A) trace on a graph paper the long-run behavior of f(x).
B) One way to find the zeros of a function (that is the x-intercepts of the graph, that is the solutions of y=0) and to predict the "look "of the short-run behavior, is to factor the function.
factor f(x) = __________________
you have ___ linear factors. The degree of the polynomial is ____.
The number of linear factor = degree of the polynomial.
C) Can you find the values of x that will cause f(x) (or y) to be zero ?
f(x) = y = 0 if x =______, x = _____, x _______
These are the ______ of f(x) or __________ of the graph or the ________ of y=0
(hint: use x-intercepts, zeros, solutions also called roots)
D) now you are ready to complete your graph. The curve crosses the x-axis for each x-intercept. Predict the bumps. (use the long run you already have). Check with your TI if you are correct.
Suppose p is a polynomial. If the formula for p has a linear factor, that is a factor of the form (x-k), then p has a zero at x = k. Conversely, if p has a zero at x=k, them p has a linear factor of the form (x-k ).
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2)f(X) = (x-4)(x2 + 4x + 3)
A)The leading term is _____, the degree is _____, the leading coefficient is _____, the maximum of x-intercepts is ______. trace the long-run behavior on a graph paper.
B) Complete the factorization f(x) = __________________
C) Find the zeros of f(x) = x-intercepts of the graphs = solutions of y = 0
D) complete your graph and check with your TI
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3) f(x) = -2x4 +8x2
A) What is the leading term ?_______, the leading coefficient _____, the max of x-intercepts ________, the degree ______, predict the long-term behavior of the graph of f(x).
B) To find out the the short term behavior, factor the function.
f(x) =_______________________
The 4 zeros of the functions are ____, _____, ______, _______
They are the x-intercepts of the graph and the values of x that cause y =0. (that is the solutions of y=0)
0 is called a multiple zero because the factor x contributing to y=0 is repeated more than once.
Multiple zeros have very special effects on the behavior of the graph in their neighborhood.
If the exponent of the factor is even, the graph bounces off the x-axis as soon as it touches it (at the multiple zero)
Can you then predict the graph of f(x) ? knowing that the graph bounces off the x-axis at x=0 ?
C) check with your TI
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4) f(x) = x2 (x-2)3
A) leading term_____, degree _____, leading coefficient _____, long-run behavior of f(x) .
B) Find the zeros of f(x). (the values of x that will cause f(x) to be zero).
x = ___ x = ___ x = ___ x= ____x = _____
x = ____ is a multiple zero and its multiplicity is 2. You know that the graph should bounce off the x-axis at that point.
x = ____ is a multiple zero and its multiplicity is 3. The exponent of the factor (x-2) is odd, that means that the graph crosses the x-axis at x=2 , but it looks flattened there.
C) predict the graph of f(x) and check using your TI.
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5) To predict the graph of a function, you need to find the leading coefficient and decide of the long run behavior. Then find the zeros and their multiplicity.
Consider:
f(x) = (x+3)2(x-2)(x-4)2
g(x) = (x+1)(x-2)3
h(x) = x(x-3)2
i(x) = x2(x-3)
j(x) = x3(x-2)4
k(x) = (x+6)(x+3)(x-2)(x-1)
Predict the graphs of the function. check with your TI
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7) Find the 3rd degree polynomial satisfying the following condition: It has a non-repeated root at x= -2 and a root which is repeated twice at x=4. Its graph passes through the point (1, -2).
(hint: f(x) should be in the form f(x) = k (x-x1)p(x-x2)n k is the stretch factor)
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8) you can also use the graph to predict the expression of the function.
predict the expression of the following functions : (see hint below)
hint: For graph1. f(x) = k (x-x1)p(x-x2)n
There are only 2 x-intercepts.
x1 = _____ and
since the curve just crosses the x-axis p = ___ (single one)
x2 = _______ and since the curve flattens out at x2, x2 is a multiple zero and n is odd.
n= 1 or n= 3 or n= 5 .. )
so f(x) = k (x-x1)p(x-x2)n = __________ (replace x1, x2, p and n by their values).
(there are different possibilities for f(x) depending on the n you chose: 1, 3, 5 .. All of them can work. we usually take the smallest odd number as a multiplicity)
Then you need to find k using the fact that (0,5) belongs to the graph. that is f(0) = 5
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PART II: Assignments from the book.
1) so p. 394 12, 16, 17, 23 Give a possible formula for the polynomials. (observe the curves)
2) p. 395 # 30
Find a possible formula for the polynomial with the following properties:
f(-3)=0 , f(1) =0, f(4)=0, and f(2) =5
3) p. 395 # 31
g is fourth degree, g has a double zero at x=3, g(5) =0, g(-1) =0 , and g(0)=3
4) #32
Least possible degree through the points (-3,0), (1,0#34) and (0,-3)
5)#34, 36, 37
Find the real zeros (if any) of the polynomials
y = x2 + 5x + 6 (hint: factor)
y=4x2 - 1 (remember ? a2 - b2 = (a-b) ( a+b)
y= 4x2 + 1
6) # 40
Let x denote the length of the side of each cut-out square. Assume negligible thickness.
a) Find a formula for the volume of the box as a function of x.
(hint: volume = length x width x height)
b) For what values of x does the formula from part (a) make sense in the context of the problem ?
c) Sketch a graph of the volume function. (use the leading term)
d) Use your TI to graph the function. use the window: -2<x<-5 and -100<y<100.
What, approximately, is the maximum volume of the box (use your TI, calc/max)
7) using the previous problem do 42.
Take a 8.5 by 11 inch piece of paper and cut out four equal squares (x) from the corners. Fold up the sides to create an open box. Find the dimensions of the box that has maximum volume.
hint: first write V(x) the volume of the box. Then trace it with the TI. Find the value of x for which v(x) is max. Find the dimension. (l=11-2x, L = 8.5 - 2x, H = x)
8) #43
Consider the function a(x)=x5 + 2x3 - 4x
a) Without using a calculator or computer, what can you say about the graph of a ? (long run)
b) Use a calculator to determine the zeros of this function to three decimal places.
hint: trace the graph using zoom 6 and them zoom box. Use calc to find the zeros.
c) Explain why you think that you have all the possible zeros..
D) What are the zeros of 2x5 + 4x3 - 8x ? Does your answer surprise you ?
1) consider the functions:
f(x) = x4 - 4x3 + 16x -16 g(x) = x4 -4x3 + 16x h(x) = x4+ x3 - 8x2 - 12x
These functions are ______ degree polynomial, the leading term is ____, the leading coefficient
is _____, the maximum number of x-intercepts is ______, the long-run behavior is:
Enter the functions in your TI as Y1, Y2, Y3. Set the window as : -10<x<10 and -200<y<200
trace the graphs. The 3 graphs look the same. right ?
What happens if you change a smaller scale. Use Zoom zbox to frame the area that includes the x-intercepts. (that is about -10<x<10 and -40<y<40). Do they still look alike ?
The 3 behaviors differ now. The intercepts are not the same so the bumps don't look alike. We say the long-run is the same because the leading term is the same but not the short-run. To study the short run, we need to study the area around the x-intercepts.
2) y = f(x) = x3 - x2 - 6x
A) trace on a graph paper the long-run behavior of f(x).
B) One way to find the zeros of a function (that is the x-intercepts of the graph, that is the solutions of y=0) and to predict the "look "of the short-run behavior, is to factor the function.
factor f(x) = __________________
you have ___ linear factors. The degree of the polynomial is ____.
The number of linear factor = degree of the polynomial.
C) Can you find the values of x that will cause f(x) (or y) to be zero ?
f(x) = y = 0 if x =______, x = _____, x _______
These are the ______ of f(x) or __________ of the graph or the ________ of y=0
(hint: use x-intercepts, zeros, solutions also called roots)
D) now you are ready to complete your graph. The curve crosses the x-axis for each x-intercept. Predict the bumps. (use the long run you already have). Check with your TI if you are correct.
Suppose p is a polynomial. If the formula for p has a linear factor, that is a factor of the form (x-k), then p has a zero at x = k. Conversely, if p has a zero at x=k, them p has a linear factor of the form (x-k ).
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2)f(X) = (x-4)(x2 + 4x + 3)
A)The leading term is _____, the degree is _____, the leading coefficient is _____, the maximum of x-intercepts is ______. trace the long-run behavior on a graph paper.
B) Complete the factorization f(x) = __________________
C) Find the zeros of f(x) = x-intercepts of the graphs = solutions of y = 0
D) complete your graph and check with your TI
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3) f(x) = -2x4 +8x2
A) What is the leading term ?_______, the leading coefficient _____, the max of x-intercepts ________, the degree ______, predict the long-term behavior of the graph of f(x).
B) To find out the the short term behavior, factor the function.
f(x) =_______________________
The 4 zeros of the functions are ____, _____, ______, _______
They are the x-intercepts of the graph and the values of x that cause y =0. (that is the solutions of y=0)
0 is called a multiple zero because the factor x contributing to y=0 is repeated more than once.
Multiple zeros have very special effects on the behavior of the graph in their neighborhood.
If the exponent of the factor is even, the graph bounces off the x-axis as soon as it touches it (at the multiple zero)
Can you then predict the graph of f(x) ? knowing that the graph bounces off the x-axis at x=0 ?
C) check with your TI
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4) f(x) = x2 (x-2)3
A) leading term_____, degree _____, leading coefficient _____, long-run behavior of f(x) .
B) Find the zeros of f(x). (the values of x that will cause f(x) to be zero).
x = ___ x = ___ x = ___ x= ____x = _____
x = ____ is a multiple zero and its multiplicity is 2. You know that the graph should bounce off the x-axis at that point.
x = ____ is a multiple zero and its multiplicity is 3. The exponent of the factor (x-2) is odd, that means that the graph crosses the x-axis at x=2 , but it looks flattened there.
C) predict the graph of f(x) and check using your TI.
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5) To predict the graph of a function, you need to find the leading coefficient and decide of the long run behavior. Then find the zeros and their multiplicity.
Consider:
f(x) = (x+3)2(x-2)(x-4)2
g(x) = (x+1)(x-2)3
h(x) = x(x-3)2
i(x) = x2(x-3)
j(x) = x3(x-2)4
k(x) = (x+6)(x+3)(x-2)(x-1)
Predict the graphs of the function. check with your TI
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7) Find the 3rd degree polynomial satisfying the following condition: It has a non-repeated root at x= -2 and a root which is repeated twice at x=4. Its graph passes through the point (1, -2).
(hint: f(x) should be in the form f(x) = k (x-x1)p(x-x2)n k is the stretch factor)
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8) you can also use the graph to predict the expression of the function.
predict the expression of the following functions : (see hint below)
 graph1. please note that it flattens out at x=2 and that the curve go through (0,5) |  graph2: note that the graph goes through (-1,-2) |  graph3: note that the bounces off the x-axis at x=-2 and that it goes through (2, -4) |
x2 = _______ and since the curve flattens out at x2, x2 is a multiple zero and n is odd.
n= 1 or n= 3 or n= 5 .. )
so f(x) = k (x-x1)p(x-x2)n = __________ (replace x1, x2, p and n by their values).
(there are different possibilities for f(x) depending on the n you chose: 1, 3, 5 .. All of them can work. we usually take the smallest odd number as a multiplicity)
Then you need to find k using the fact that (0,5) belongs to the graph. that is f(0) = 5
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PART II: Assignments from the book.
1) so p. 394 12, 16, 17, 23 Give a possible formula for the polynomials. (observe the curves)
2) p. 395 # 30
Find a possible formula for the polynomial with the following properties:
f(-3)=0 , f(1) =0, f(4)=0, and f(2) =5
3) p. 395 # 31
g is fourth degree, g has a double zero at x=3, g(5) =0, g(-1) =0 , and g(0)=3
4) #32
Least possible degree through the points (-3,0), (1,0#34) and (0,-3)
5)#34, 36, 37
Find the real zeros (if any) of the polynomials
y = x2 + 5x + 6 (hint: factor)
y=4x2 - 1 (remember ? a2 - b2 = (a-b) ( a+b)
y= 4x2 + 1
6) # 40
Let x denote the length of the side of each cut-out square. Assume negligible thickness.
a) Find a formula for the volume of the box as a function of x.
(hint: volume = length x width x height)
b) For what values of x does the formula from part (a) make sense in the context of the problem ?
c) Sketch a graph of the volume function. (use the leading term)
d) Use your TI to graph the function. use the window: -2<x<-5 and -100<y<100.
What, approximately, is the maximum volume of the box (use your TI, calc/max)
7) using the previous problem do 42.
Take a 8.5 by 11 inch piece of paper and cut out four equal squares (x) from the corners. Fold up the sides to create an open box. Find the dimensions of the box that has maximum volume.
hint: first write V(x) the volume of the box. Then trace it with the TI. Find the value of x for which v(x) is max. Find the dimension. (l=11-2x, L = 8.5 - 2x, H = x)
8) #43
Consider the function a(x)=x5 + 2x3 - 4x
a) Without using a calculator or computer, what can you say about the graph of a ? (long run)
b) Use a calculator to determine the zeros of this function to three decimal places.
hint: trace the graph using zoom 6 and them zoom box. Use calc to find the zeros.
c) Explain why you think that you have all the possible zeros..
D) What are the zeros of 2x5 + 4x3 - 8x ? Does your answer surprise you ?
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