advanced functions questions

please indicate which question you are answering, and thanks for your help

1. Solve the following equation:

(x+1)^2 - 3(x+1) = 0

2. What is the maximum possible number of turning points for each of the following functions?

a) linear function
b) quadratic function
c) cubic function
d) quartic function

3. For the function f(x) = (x-4)(x+1)(x-1)
a) find the domain and range
b) find the real zeros, to the nearest tenth necessary
c) the y-intercepts
d) the end behaviour (what is this?)
e) any symmetry
f) the number of turning points

Quote:

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Originally Posted by juiicycouture

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1. Solve the following equation:

(x+1)^2 - 3(x+1) = 0

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Factor the term:
(x+1)^2 - 3(x+1) = (x+1)((x+1) - 3) = (x+1)(x-2) = 0

Therefore the solutions are: x = -1 or x = 2

Quote:

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Originally Posted by juiicycouture

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2. What is the maximum possible number of turning points for each of the following functions?

a) linear function
b) quadratic function
c) cubic function
d) quartic function

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If n is the degree (or do you say grade?) of the function then the maximum number of turning points is n-1:

Code:

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prob  degree  max. number of turning points
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a)        1                        0
b)        2                        1 (= vertex)
c)        3                        2
d)        4                        3

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Quote:

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Originally Posted by juiicycouture

3. For the function f(x) = (x-4)(x+1)(x-1)
a) find the domain and range
b) find the real zeros, to the nearest tenth necessary
c) the y-intercepts
d) the end behaviour (what is this?)
e) any symmetry
f) the number of turning points

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a) There aren't any restrictions for x thus d = IR (the set of real numbers).
Because f has the degree 3 and the coefficient of x³ is 1 the values of the function reach from -∞ to +∞

b) f(x) = 0 if one of the factors is zero. Therefore the zeros are: x = -1, x = 1, x = 4

c) The y-intercept is f(0) = (-4)(1)(-1) = 4

d) If x --> -∞ then f(x) --> -∞
If x --> ∞ then f(x) --> ∞

e) Because f has the degree 3 there is only possible a point inflection. The centre of symmetry could only be the inflection point. With your function the graph of f is symmetric to the point ((4/3), -(56/27))

f) see #2: thus 2

EB

Quote:

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Originally Posted by juiicycouture

please indicate which question you are answering, and thanks for your help

1. Solve the following equation:

(x+1)^2 - 3(x+1) = 0

2. What is the maximum possible number of turning points for each of the following functions?

a) linear function
b) quadratic function
c) cubic function
d) quartic function

3. For the function f(x) = (x-4)(x+1)(x-1)
a) find the domain and range
b) find the real zeros, to the nearest tenth necessary
c) the y-intercepts
d) the end behaviour (what is this?)
e) any symmetry
f) the number of turning points

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i would approach these by graphing, the simplest way to figure them out. is this MHF4U work?

2.
a) no turning points, straight line
b) one turning point, a simple parabola
c) two turning points
d) three turning points