Functions: Unit 3 Review
1. State the end behaviours of each of the following functions.
a)
b)
c)
d)
f ( x)  2 x 4  3x 3  2 x  3
f ( x)  5x3  x 2  3x  5
f ( x)  3x 5  2 x 4  3x
f ( x)   x6  2 x5  98
2. State whether each function has an even number of turning points or an odd number of turning points.
a) f ( x)  4 x9  2 x 2
b)
c)
d)
f ( x)  8x6  2 x 4  3x  3
f ( x)   x 7  6 x  2
f ( x)  2 x 4  5 x  3
3. Sketch a possible graph of each of the following functions.
a)
b)
c)
d)
f ( x)  2( x  3)( x  4)2
f ( x)  ( x  5)2 ( x  2)2
f ( x)  2( x  1)2 ( x  3)2 ( x  6)
f ( x)  3( x  4)3 ( x  4)
4. Describe the transformations that were applied to y  x5 to get the following functions
.
a) y  3(2( x  1))5  4
5
b)
1

y  2  ( x  3)   5
2

5. Sketch the graph of a polynomial function that satisfies the following set of conditions:
a) Degree 3, negative leading coefficient, 2 zeros, 2 turning points.
b) Degree 4, positive leading coefficient, 2 zeros, 3 turning points
6. Determine if each of the following functions is even, odd, or neither even nor odd.
a) f ( x)  3x3  x
b)
c)
f ( x)  3 x 4  2 x 2
f ( x)  x 3  4 x  2
7. Give an example of a polynomial function that has no absolute maximum or minimum.
8. Give an example of a polynomial function that has an absolute maximum.
9. Give an example of a polynomial function that has an absolute minimum
10. Given the polynomial function f ( x)  ax 4  bx3  cx 2  dx  e , what must be true about the
coefficients if f is an even function?
11. Determine the equation of the cubic function that has zeros at -2, 1, and 4 if f (3)  2 .
12. The function y  x 4 has undergone the following sets of transformations. If y  x 4 passes through the
points (-1, 1), (0, 0), and (2, 16), list the coordinates of these transformed points on each new curve.
a) Vertically compressed be a factor of
1
, horizontally stretched by a factor of 3, and horizontally
2
translated 5 units to the right.
b) Reflected in the x-axis, stretched vertically be a factor of 2, and translated 3 units up.
c) Reflected in the y-axis, horizontally compressed by a factor of
1
, and translated three units to the
4
left.
13. Determine the x-intercepts of each of the following polynomial functions. Round to two decimal places
if necessary.
a) y  ( x  3)3  5
b) y  2( x  4)4  20
14. Calculate each of the following using long division.
(2 x4  3x2  x  8)  ( x  3)
b) (2 x3  5x 2  4 x  5)  (2 x  1)
a)
15. Calculate each of the following using synthetic division.
a) (3x3  5x 2  10)  ( x  3)
b) (12 x4  56 x3  59 x 2  9 x  18)  (2 x  1)
16. Find the dividend if the divisor is 3x  2 , the quotient is x3  x  12 , and the remainder is 15.
17. Find the divisor if the dividend is 5x3  x2  3 , the quotient is 5x 2  14 x  42 , and the remainder is
123 .
18. Use the remainder theorem to determine the remainder when (2 x3  13x 2  4 x  6) is divided by x  3 .
19. The polynomial 2 x3  9 x2  kx  21 has 2 x  1as one of its factors. Determine the value of k.
20. Factor fully. State the zeros of the function. Sketch the function using the available information.
a) f ( x)  2 x3  3x 2  8 x  12
b)
c)
f ( x)  2 x3  9 x2  10 x  3
f ( x)  2 x4  x3  26 x2  37 x  12
21. Factor each expression.
a) x3  27
b) 125x3  1
e)
343x6  1
c) 5x 4  40 x
d) ( x  3)3  (2 x  1)3