MHF 4U
Exercises
1. Use the graph of each polynomial function to identify the polynomial as cubic or quartic, state the
sign of the leading coefficient of its function, describe the end behaviour, and say whether the graph
has a turning point.
(a)
(b)
(c)
Type
Sign of Leading
Coeff.
End Behaviour
Turning Points
2. Sketch the graph of a polynomial function that satisfies each set of conditions.
(a) degree 4, positive leading coefficient, 3 zeros, 3 turning points
(b) degree 4, negative leading coefficient, 2 zeros, 1 turning point
(c) degree 4, positive leading coefficient, 1 zero, 3 turning points
(d) degree 3, negative leading coefficient, 1 zero, no turning point
(e) degree 3, positive leading coefficient, 2 zeros, 2 turning points
Solution
(a)
(b)
(d)
MHF 4U
(c)
(d)
(e)
3. Check Your Understanding: Copy and complete the table.
Degree of
Sign of Leading
End Behaviour of ƒ(x)
End Behaviour of ƒ(x)
ƒ(x)
Coefficient of ƒ(x)
as
as
odd
positive
even
negative
odd
negative
even
positive
MHF 4U
Exercises continued
4. Identify the function that corresponds to each graph. Justify your choices.
a) g(x) = x3 – x2
b) h(x) = x4 – 3x3 + x – 1
d) j(x) = -x4 – x3 + 11x2 + 9x – 3
c) f(x) = -3x3 + 8x2 + 7
i)
ii)
iii)
iv)
5. Sketch the following:
a) y = 2x2 – 9x + 4
b) y = –2(x + 1)(x – 1)
c) y = (x – 1)(x – 3)(x + 2)
MHF 4U
6. Determine the equation of the function given:
a. A quadratic function with zeros -3 and 2 and a y-intercept of 12.
b. A cubic function with zeros -2, 1, 4 and a y-intercept of 24.
c. A cubic function with zeros -5, 3 and 0 and passes through the point (4, -2).
d. A quartic function with zeros -2, 0, 0, 1 and passes through (-3, -12).
MHF 4U
7. A cubic function has zeros -3, -1, 2. The y-intercept of its graph is 12.
a. Determine the equation of the function.
b. Sketch the graph of the function.
MHF 4U
8. Determine the equation of the function, then sketch the graph.
a. quadratic function with zero 2 (or order 2);
graph has y-intercept 12.
c. cubic function with zeros -2 and 2 (of order 2);
graph has y-intercept -16
b. cubic function with zeros -2, 1 and 4;
graph has y-intercept 24
d.
cubic function with zeros 0, 2 and 4;
graph passes through (3, 9)
MHF 4U
9. Determine the zeros of each equation.
a. f ( x )  x 2  10 x  16
b. f ( x )  x 3  7 x 2  12 x
10. Determine the equation of each cubic function.
a)
b)
c. f ( x )  x 3  5x 2  14 x
MHF 4U
11. Determine an equation to represent the graph of each polynomial functions.
a)
b)
c)
d)
MHF 4U
Warm Up!
Match each graph with the appropriate polynomial function.
(a)
(b)
(c)
(d)
i. ƒ(x) = (x + 3)(x – 2)2(x + 1)
iii. ƒ(x) = – (x – 1)2(x + 3)
ii. ƒ(x) = (x – 1)(x – 4)(x + 4)
iv. ƒ(x) = (x – 2)2(x + 3)2