Pre-Calculus 12
Unit 3.1 Assignment.
Name: _____________________
1. Determine whether each function is a polynomial function. Justify your answers.
a) f (x) = 2x4 – 3x + 2
The degree is ______, which is an _____________ number. f (x) _______ a polynomial
function.
x
b) y = 3 + 5
x
The term 3 means this is an ______________ function. This function _________ a
polynomial function.
c) g (x) = 9
g (x) has degree ___________. This function ___________ a polynomial function.
d) f (x) = x–2 +7x3+ 1
–2
The term x has a degree, which ______________ a natural number. f (x) _______ a
polynomial function.
2. Complete the table for each polynomial function.
3. For each graph of a polynomial function, fill in the blank.
a) f (x) = –x3 – x2 + 5x – 3
The graph extends from quadrant ______ to quadrant _______.
The function has _________ degree.
The leading coefficient is __________.
There are _______ x-intercepts.
Domain: ______________ , Range: ______________
b)
The graph extends from quadrant ______ to quadrant _______.
The function has _________ degree.
The leading coefficient is __________.
There are _______ x-intercepts.
Domain: ______________ , Range: ______________
c)
The graph extends from quadrant ______ to quadrant _______.
The function has _________ degree.
The leading coefficient is __________.
There are _______ x-intercepts.
Domain: ______________ , Range: ______________
4. For each function, use the degree and the sign of the leading coefficient to describe the end behavior
of its graph. State the possible number of x-intercepts and the value of the y-intercept.
a) g (x) = 4x5 – x3 + 3x2 – 6x + 2
The degree is ________ with a _________ as leading coefficient.
The graph extends from _________ quadrant to ________ quadrant.
There are a maximum of _______ x-intercepts. The y-intercept is ________.
b) y = –x4 – 2x5 + x3 – 3x2 + x
The degree is ________ with a _________ as leading coefficient.
The graph extends from _________ quadrant to ________ quadrant.
There are a maximum of _______ x-intercepts. The y-intercept is ________.
c) h(x) = x – 7x3 – 6
The degree is ________ with a _________ as leading coefficient.
The graph extends from _________ quadrant to ________ quadrant.
There are a maximum of _______ x-intercepts. The y-intercept is ________.
5. On each set of axes, sketch a polynomial function with the given characteristics.
a) A polynomial function with degree 3,
a positive leading coefficient, and
2 x-intercepts.
b) A polynomial function with degree 4,
a negative leading coefficient, and
4 x-intercepts.
c) A polynomial function with degree 5,
a negative leading coefficient, and
3 x-intercepts.
d) A polynomial function with degree 4,
a positive leading coefficient, and
2 x-intercepts.
6. A snowboard manufacturer determines that its profit, P, in dollars, can be modeled by the function
4
P(x) = 1000x + x – 3000, where x represents the number, in hundreds, of snowboards sold.
a) What is the degree of the function P(x)?
b) What are the leading coefficient and constant of this function? What does the constant
represent?
c) Describe the end behavior of the graph of this function.
d) What are the restrictions on the domain of this function? Explain why you selected those
restrictions.
e) What do the x-intercepts of the graph represent for this situation?
f) What is the profit from the sale of 1500 snowboards?
7. Populations in rural communities have declined in Western Canada, while populations in larger
urban centers have increased. This is partly due to expanding agricultural operations and fewer
traditional family farms. A demographer uses a polynomial function to predict the population, P,
of a town t years from now. The function is P(t) = t 4 – 20t3 – 20t2 + 1500t + 15 000. Assume this
model can be used for the next 20 years.
a) What are the key features of the graph of this function?
b) What is the current population of this town?
c) What will the population of the town be 10 years from now?
d) When will the population of the town be approximately 24 000?