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2.08 Graphs of cubics

Worksheet
The shape of a cubic
1

Consider the graph of y = x^{3}.

a

As x becomes larger in the positive direction (ie x approaches infinity), what happens to the corresponding y-values?

b

As x becomes larger in the negative direction (ie x approaches negative infinity), what happens to the corresponding y-values?

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

Consider the given graph of a cubic function.

a

Determine whether the cubic is positive or negative.

b

State the coordinates of the y-intercept.

c

State the equation of the function.

-3
-2
-1
1
2
3
x
-3
-2
-1
1
2
3
4
5
6
7
y
3

Consider the cubic function y = \dfrac{1}{2} x^{3} + x.

a

Determine whether the cubic is positive or negative.

b

Sketch the graph of y = \dfrac{1}{2} x^{3} + x.

c

State the coordinates of the x-intercept.

4

Consider the graph of the function y = x^{3}.

Determine the point where the curve changes from being concave down to being concave up.

This is called the point of inflection.

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

Consider the graph of the function y = - x^{3}.

Out of the points A, B and C:

a

At which point is the curve concave up?

b

Which is the point of inflection?

c

At which point is the curve concave down?

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

For each of the following functions:

i

For what values of x is the cubic concave up?

ii

For what values of x is the cubic concave down?

iii

State the coordinates of the point of inflection.

a
-3
-2
-1
1
2
3
x
-6
-4
-2
2
4
6
y
b
1
2
3
4
5
x
-2
-1
1
2
3
4
5
6
y
7

A cubic function is defined as y = \dfrac{1}{2} x^{3} + 4.

a

Find the x-intercept of the function.

b

Find the y-intercept of the function.

8

Consider the graph of the function:

a

The equation of the function can be written as y = a x^{3} + b x^{2} + c x + d.

Is the value of a is positive or negative?

b

State the coordinates of the y-intercept.

c

For which values of x is the graph concave up?

d

For which values of x is the graph concave down?

e

State the coordinates of the point of inflection.

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

Consider the graph of the cubic function shown.

For what values of x is y \geq 0?

-3
-2
-1
1
2
3
x
-20
-15
-10
-5
5
10
15
20
y
Transformations
10

Determine whether the following statements are true of the graphs of y = \dfrac{1}{2} x^{3} and y = x^{3}.

a

One is a reflection of the other about the y-axis.

b

y increases more rapidly on y = \dfrac{1}{2} x^{3} than on y = x^{3}.

c

y = \dfrac{1}{2} x^{3} is a horizontal shift of y = x^{3}.

d

y increases more slowly on y = \dfrac{1}{2} x^{3} than on y = x^{3}.

11

The graph of y = x^{3} has a point of inflection at \left(0, 0\right). By considering the transformations that have taken place, find the point of inflection of each cubic curve below:

a

y = \dfrac{2}{3} x^{3}

b

y = x^{3} + 3

c

y = - x^{3} + 4

12

For each cubic function below:

i

Complete a table of values of the form:

x-2-1012
y
ii

Sketch the graph.

a
y = x^{3}
b
y = x^{3} - 2
c
y = - x^{3} + 5
13

Consider the curve y = x^{3} - 8.

a

Find the x-intercept.

b

Find the y-intercept.

c

Find the horizontal point of inflection.

d

Sketch the graph of the curve.

14

Consider the curve y = 3 x^{3} + 3.

a

Find the x-intercept.

b

Find the y-intercept.

c

State the coordinates of the point of inflection.

d

Sketch the graph of the curve.

15

Consider the equation y = x^{3} - 3.

a

Complete the following set of points for the given equation.

A(-3, ⬚), B(-2, ⬚), C(-1, ⬚), D(0, ⬚), E(1, ⬚), F(2, ⬚), G(3, ⬚)

b

Sketch the curve that results from the entire set of solutions for the equation being graphed.

16

Consider the cubic function y = x^{3} - 4.

a

State the y-intercept of the function.

b

Complete the following table of values.

x-112
y
c

Find the domain of the function in interval notation.

d

Find the range of the function in interval notation.

e

Sketch the graph.

17

Bianca completed the table of values for the equation y = 20 - x^{3}.

x-6-4-2024
y236842820-4-44
a

One of the points in the table is incorrect. Which point is it?

b

Bianca wants to find one other pair of values that satisfy y = 20 - x^{3} before graphing the curve. Find the ordered pair when the x-coordinate is 6.

c

Plot the complete set of solutions for y = 20 - x^{3}, making sure that the curve goes through all points that satisfy it.

18

Consider the curve y = - 2 x^{3} + 16.

a

Find the x-intercept.

b

Find the y-intercept.

c

State the coordinates of the point of inflection.

d

Find the domain of the function in interval notation.

e

Find the range of the function in interval notation.

f

Sketch the graph of the curve.

19

A graph of f(x) = x^{3} is shown. Sketch the curve after it has undergone transformations resulting in the function g(x) = x^{3} - 4.

-3
-2
-1
1
2
3
x
-6
-4
-2
2
4
6
y
20

A graph of y = x^{3} is shown. Sketch the curve after it has undergone transformations resulting in the function y = 2 \left(x - 2\right)^{3} - 2.

-4
-3
-2
-1
1
2
3
4
x
-6
-4
-2
2
y
21

For the following cubic functions:

i

Determine whether the cubic is increasing or decreasing from left to right.

ii

Determine whether the cubic is more or less steep than the cubic y = x^{3}.

iii

Find the coordinates of the point of inflection of the cubic.

iv

Sketch the graph.

a

y = 2 x^{3} + 2

b

y = 4 x^{3} - 3

c

y = - \dfrac{x^{3}}{4} + 2

22

Consider the graph of y = x^{3} shown:

a

How do we shift the graph of y = x^{3} to get the graph of y = \left(x - 2\right)^{3} - 3 ?

b

Hence, sketch y = \left(x - 2\right)^{3} - 3.

-3
-2
-1
1
2
3
x
-6
-4
-2
2
4
6
y
23

Consider the function y = 2 \left(x - 2\right)^{3} - 2

a

Is the cubic increasing or decreasing from left to right?

b

Is the function more or less steep than the function y = x^{3} ?

c

What are the coordinates of the inflection point of the function?

d

Sketch the graph y = 2 \left(x - 2\right)^{3} - 2.

24

Consider the curve y = 2 \left(x + 1\right)^{3} + 16.

a

Find the x-intercept.

b

Find the y-intercept.

c

State the coordinates of the point of inflection.

d

Sketch the graph of the curve.

25

Consider the curve y = - 3 \left(x - 1\right)^{3} + 3.

a

Find the x-intercept.

b

Solve for the y-intercept.

c

State the coordinates of the point of inflection.

d

Sketch the graph of the curve.

Factored form of a cubic function
26

The cubic function y = \left(x + 1\right) \left(x - 2\right) \left(x - 3\right) has been graphed below. Determine the number of solutions to the equation \left(x + 1\right) \left(x - 2\right) \left(x - 3\right) = 0.

-2
-1
1
2
3
4
x
-4
-2
2
4
6
8
10
y
27

A cubic function has the equation y = x \left(x - 4\right) \left(x - 3\right). How many x-intercepts will it have?

28

State whether the following functions pass through the origin:

a

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

b

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

c

y = \left(x - 4\right) \left(x + 7\right) \left(x - 5\right)

d

y = x \left(x - 6\right) \left(x + 8\right)

29

State whether each function has exactly two x-intercepts:

a

y = \left(x + 3\right)^{3}

b

y = \left(x + 6\right)^{2} \left(x + 5\right)

c

y = \left(x + 7\right) \left(x - 1\right) \left(x - 4\right)

d

y = x \left(x - 2\right) \left(x - 8\right)

30

Consider the cubic function y = \left(x + 3\right) \left(x - 2\right) \left(x - 5\right).

a

Determine the x-intercepts.

b

A second cubic function has the same x-intercepts, but is a reflection of the above function about the x-axis. State the equation of the reflected function.

31

Sketch the function y = \left(x - 2\right) \left(x + 1\right) \left(x + 4\right) showing the general shape of the curve and the x-intercepts.

32

Sketch the graph of f(x) = \left(x - 2\right)^{2} \left(x - 4\right) and g(x) = - \left(x - 2\right)^{2} \left(x - 4\right) on the same number plane.

33

Consider the function y = \left(x + 3\right)^{3}.

a

Complete the following table of values:

x-5-4-3-2-1
y
b

Sketch the graph.

c

State the domain.

d

State the range.

34

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

a

Complete the following table of values:

x01234
y
b

Sketch the graph.

35

Consider the function y = \left(x + 1\right)^{3}.

a

Complete the following table of values:

x-3-2-101
y
b

Sketch the graph.

36

Consider the equation y = \left(x - 4\right)^{3}.

a

Complete the set of solutions for the above equation.

A (3,⬚) , B (2,⬚) , C(5,⬚) , D (4,⬚) , E (6,⬚)

b

Plot the points on a coordinate axes.

c

On the same axes, plot the curve that results from the entire set of solutions for the equation being graphed.

d

Consider x = 6.17. According to the points on the graph, between which two integer values should the corresponding y-value lie?

37

For each of the functions below:

i

Find the x-intercepts.

ii

Find the y-intercept.

iii

Sketch the graph.

a

y = \left(x + 3\right) \left(x + 2\right) \left(x - 2\right)

b

y = - \left(x + 4\right) \left(x + 2\right) \left(x - 1\right)

c

y = \left(x - 2\right)^{2} \left(x + 5\right)

38

For each of the functions below:

i

Express the equation in factorised form.

ii

Find the y-intercept of the graph.

iii

Find the x-intercepts of the graph.

iv

Sketch the graph of the curve.

a

y = 3 x + 2 x^{2} - x^{3}

b

y = x^{3} - 4 x^{2} - 7 x + 10

Applications
39

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

a

Fill in the following table of values for the equation V = \dfrac{4}{3} \pi r^{3}, in terms of \pi:

r12457
V
1
2
3
4
5
6
7
r
36\pi
72\pi
108\pi
144\pi
180\pi
216\pi
252\pi
288\pi
324\pi
V
b

Determine whether the following intervals show the volume, V of a sphere that has a radius measuring 4.5\text{ m}.

i
20\pi \lt V \lt 57\pi
ii
85\pi \lt V \lt 166\pi
iii
179\pi \lt V \lt 696\pi
iv
221\pi \lt V \lt 366\pi
c

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

40

A cube has side length 7\text{ cm} and a mass of 1715\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. Find the value of k.

b

Hence state the equation relating the mass (m) and side length (x) of a cube.

c

Complete the table of values:

x1234
m
d

Sketch the graph of your equation.

e

From the equation, find the mass of a cube with side 8.5\text{ cm}, to the nearest gram.

f

A cube has a mass of 1920\text{ g}. From your graph, determine what whole number value its side length is closest to.

41

A box without a top cover is to be constructed from a rectangular cardboard, measuring 6\text{ cm} by 10\text{ cm} by cutting out four square corners of length x\text{ cm}. Let V represent the volume of the box.

a

Express the volume V of the box in terms of x, writing the equation in factorised form.

b

For what range of values of x is the volume function defined?

c

Sketch the graph of the volume function.

d

Determine the volume of a box that has a height equivalent to the shorter dimension of the base.

42

A cylinder of radius x and height 2 h is inscribed in a sphere of radius \sqrt{15}, centre at O as shown:

a

Form an equation relating x and h.

b

Form an expression for V, the volume of the cylinder, in terms of h.

c

Determine the domain of h.

d

The cylinder with the largest possible volume has a height of 2 \sqrt{5}, so h = \sqrt{5}.

Determine the exact volume of this cylinder.

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Outcomes

ACMMM017

recognise features of the graphs of y=x^3, y=a(x−b)^3+c and y=k(x−a)(x−b)(x−c), including shape, intercepts and behaviour as x→∞ and x→−∞

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