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Stage 5.1-3

4.10 Sketching polynomials

Worksheet
Graphs of polynomials
1

For each of the following graphs of the function y = f \left( x \right):

i

Write the x-values of the x-intercepts.

ii

Write the y-value of the y-intercept.

iii

Determine the behaviour of the function as x \to \infty.

iv

Determine the behaviour of the function as x \to -\infty.

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
b
-4
-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-4
-2
2
4
6
8
10
12
14
16
y
e
-3
-2
-1
1
2
3
x
-1
1
2
3
4
5
y
f
-3
-2
-1
1
2
3
4
5
6
7
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
2

Consider the function f \left( x \right) = x^{4}.

a

Complete the following table of values:

x-4-2-10124
f(x)
b

Hence, sketch the curve.

3

Consider the cubic function f \left( x \right) = 2 x^{3} - 7.

a

Complete the following table of values:

x-5-1015
f(x)
b

The function appears to be asymmetric about which point?

c

At which x-value would the function have a value between - 9 and - 257?

d

Does the function increase or decrease for values of x > 0?

e

Hence, sketch the curve.

4

Consider the cubic function f \left( x \right) = 4 x^{3} + 8 x^{2}.

a

Complete the following table of values:

x-8-3-1017
f(x)
b

Hence, sketch the curve y = f \left( x \right).

c

State the x-values of the x-intercepts.

d

Find the function value at x = 2.

e

Does the function increase or decrease for values of x > 0?

5

Consider P \left( x \right) = x^{3} + 2 x^{2} - 5 x - 6 shown:

a

State all the possible rational zeros.

b

Determine the actual zeros of P \left( x \right).

c

Hence, factorise P \left( x \right) .

-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
6

Consider P \left( x \right) = x^{3} - 3 x^{2} - 6 x + 8 shown:

a

State all the possible rational zeros.

b

Determine the actual zeros of P \left( x \right).

c

Hence, factorise P \left( x \right) .

-5
-4
-3
-2
-1
1
2
3
4
5
x
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
y
7

Consider P \left( x \right) = x^{3} - 5 x^{2} - 29 x + 105 shown:

a

State all the possible integer zeros.

b

Determine the actual zeros of P \left( x \right).

c

Hence, factorise P \left( x \right) .

-10
-8
-6
-4
-2
2
4
6
8
10
x
-60
-40
-20
20
40
60
80
100
120
140
y
8

Consider P \left( x \right) = x^{3} - 4 x^{2} + x + 6 shown:

a

State all the possible rational zeros.

b

Determine the actual zeros of P \left( x \right).

c

Hence, factorise P \left( x \right) .

-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
9

A polynomial has been graphed with each of its intercepts shown:

a

Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:

i

y = k \left(x + 6\right)^{2} \left(x + 2\right) \left(x - 4\right)

ii

y = k \left(x - 6\right)^{2} \left(x - 2\right) \left(x + 4\right)

iii

y = k \left(x - 6\right) \left(x - 2\right) \left(x + 4\right)

iv

y = k \left(x + 6\right) \left(x + 2\right) \left(x - 4\right)

b

Determine the value of k.

c

Hence, state the equation of the least degree polynomial for the displayed graph.

-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
x
-8
-6
-4
-2
2
4
6
8
10
12
14
16
18
y
10

A polynomial has been graphed with each of its intercepts shown:

a

Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:

i

y = k \left(x - 1\right)^{2} \left(x + 2\right)

ii

y = k \left(x + 1\right)^{2} \left(x - 2\right)

iii

y = k \left(x + 1\right)^{3} \left(x - 2\right)

iv

y = k \left(x - 1\right)^{3} \left(x + 2\right)

b

Determine the value of k.

c

Hence, state the equation of the least degree polynomial for the displayed graph.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-8
-6
-4
-2
2
4
6
8
10
y
11

A polynomial has been graphed with each of its intercepts shown.

a

Let k be a non-zero real number. Determine whether the following polynomials could describe the graph:

i

y = k \left(x + 2\right)^{2} \left(x - 1\right) \left(x - 4\right)^{2}

ii

y = k \left(x + 2\right) \left(x - 1\right)^{3} \left(x - 4\right)

iii

y = k \left(x - 2\right)^{2} \left(x + 1\right) \left(x + 4\right)^{2}

iv

y = k \left(x - 2\right)^{2} \left(x + 1\right)^{3} \left(x + 4\right)^{2}

b

Determine the value of k.

c

Hence, state the equation of the least degree polynomial for the displayed graph.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-8
-6
-4
-2
2
4
6
8
10
y
Applications
12

The volume of a sphere has the formula V = \dfrac{4}{3} \pi r^{3}. The graph relating r and V is shown below.

a

Complete the table of values, correct to two decimal places.

r12457
V
b

A sphere has radius measuring 3.5\text{ m}. Determine the exact interval that the volume of the sphere falls within.

c

According to the graph, what is the radius of a sphere of volume 288 \pi \text{ m}^3?

1
2
3
4
5
6
7
r \text{ (metres)}
36\pi
72\pi
108\pi
144\pi
180\pi
216\pi
252\pi
288\pi
324\pi
V\text{ (cubic metres)}
13

A cube has side length 5\text{ cm} and a mass of 375\text{ g}. The mass of the cube is directly proportional to the cube of its side length.

a

Let k be the constant of proportionality for the relationship between the side length, x, and the mass, m, of a cube, such that m=kx^3. Solve for k.

b

Hence state the equation relating the mass and side length of the cube.

c

Sketch the curve of for this equation.

d

Find the mass of a cube with side 6.5\text{ cm}, to the nearest gram.

e

A cube has a mass of 576\text{ g}. Determine what whole number value its side length is closest to.

14

The mass in grams, M, of a cube of cork is given by the formula M = 0.28 l^{3}, where l is the side length of the cube in centimetres.

a

What is the mass of a cubic centimetre of cork to the nearest 0.01 of a gram?

b

Complete the table values, to the nearest 0.01 of a gram if necessary.

\text{Side length(cm)}1234567
\text{Mass (g)}
c

Sketch the graph that represents the function.

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Outcomes

MA5.3-9NA

sketches and interprets a variety of non-linear relationships

MA5.3-10NA

recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems

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