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iGCSE (2021 Edition)

5.06 Sketching polynomials

Worksheet
Shape of cubics
1

Is the function y = - 2 x^{3} - 4 one-to-one?

2

By considering the graph of y = x^{3}, determine the following:

a

As x approaches infinity, what happens to the corresponding y-values?

b

As x approaches negative infinity, what happens to the corresponding y-values?

-4
-3
-2
-1
1
2
3
4
x
-15
-10
-5
5
10
15
y
3

Determine whether the graphed function shown has an even or odd power. Explain your answer.

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

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

a

For what values of x is the functions concave down?

b

For what values of x is the functions concave up?

c

At what point does the concavity of the curve change?

d

What is this point called?

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

Consider the graph of the function:

a

For what values of x is the cubic concave up?

b

For what values of x is the cubic concave down?

c

State the coordinates of the point of inflection.

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

How does the graph of y = \dfrac {1}{2} x^{3} differ to the graph of y = x^{3}?

7

Consider the graph of the cubic function shown. For which values of x is y \geq 0?

-3
-2
-1
1
2
3
4
x
-15
-10
-5
5
10
15
y
8

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

Finding intercepts and equations of cubics
9

Consider the graph of the function with equation of the form y = a x^{3} + b x^{2} + c x + d:

a

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
10

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)

11

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.

12

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

13

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

Which of the following could be the equation of the function?

A
y = - 2 x^{3} + 3
B
y = 2 x^{3} + 3
C
y = 2 x^{3} - 3
D
y = - 2 x^{3} - 3
-2
-1
1
2
x
-2
-1
1
2
3
4
y
14

If the graph of y = x^{3} is moved to the right by 10 units, what is the new equation?

Graphs of cubics
15

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

a

Complete the following table of values.

x-2-1012
y
b

Sketch the graph of the curve.

c

Sketch the graph of y=\vert x^3 \vert.

16

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.

17

For each of the functions below:

i

Is the cubic increasing or decreasing from left to right?

ii

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

iii

What are the coordinates of the point of inflection?

iv

Sketch the graph.

a

y = 2 x^{3} + 2

b

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

c

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

d

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

18

Consider the function f(x)=\left(x - 2\right) \left(x + 1\right) \left(x + 4\right). Sketch the following functions showing the general shape of the curve and the x-intercepts:

a

y = f(x)

b

y = \vert f(x) \vert

19

For each of the following curves:

i

Find the x-intercept(s).

ii

Find the y-intercept(s).

iii

Sketch the graph of the curve.

a

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

b

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

c

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

d

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

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Outcomes

0606C3.4

Sketch the graphs of cubic polynomials and their moduli, when given in factorised form y = k(x – a)(x – b)(x – c).

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