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) / (d)
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)


(c) (d) (e)

3. Check Your Understanding: Copy and complete the table.

Degree of
ƒ(x) / Sign of Leading
Coefficient of ƒ(x) / End Behaviour of ƒ(x)
as / End Behaviour of ƒ(x)
as
odd / positive
even / negative
odd / negative
even / positive


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

c) f(x) = -3x3 + 8x2 + 7 d) j(x) = -x4 – x3 + 11x2 + 9x – 3

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)


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).

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.

8. Determine the equation of the function, then sketch the graph.

a.  quadratic function with zero 2 (or order 2); b. cubic function with zeros -2, 1 and 4;

graph has y-intercept 12. graph has y-intercept 24

c. cubic function with zeros -2 and 2 (of order 2); d. cubic function with zeros 0, 2 and 4;

graph has y-intercept -16 graph passes through (3, 9)

9. Determine the zeros of each equation.

a. b. c.

10. Determine the equation of each cubic function.

a) b)

11. Determine an equation to represent the graph of each polynomial functions.

a) b)

c) d)

Warm Up!

Match each graph with the appropriate polynomial function.

(a) (b)

(c) (d)

i. ƒ(x) = (x + 3)(x – 2)2(x + 1) ii. ƒ(x) = (x – 1)(x – 4)(x + 4)

iii. ƒ(x) = – (x – 1)2(x + 3) iv. ƒ(x) = (x – 2)2(x + 3)2