Topics
10. Differentiation
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Standard level

10.09 Graphs and the derivative

Lesson
Worksheet
Practice
Worksheet
Functions and gradient functions
1

Consider the graph of the function \\ f \left( x \right) = - \left(x - 4\right)^{3} + 7:

a

State the x-value of the stationary point of f \left( x \right).

b

State the domain where f \left( x \right) is decreasing.

1
2
3
4
5
6
7
8
x
1
2
3
4
5
6
7
8
9
10
11
f (x)
2

State whether the rate of change of the following functions is positive, negative, or zero for all values of x:

a
f \left( x \right) = - 8
b
f \left( x \right) = 4 x - 1
c
f \left( x \right) = - x + 10
3

For each of the following functions:

i

State the x-value of the stationary point of f \left( x \right).

ii

State the domain where f \left( x \right) is increasing.

iii

State the domain where f \left( x \right) is decreasing.

a

f \left( x \right) = - 2 \left(x + 2\right)^{2} + 4

-5
-4
-3
-2
-1
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
f (x)
b

f \left( x \right) = \left(x + 4\right) \left(x - 2\right)

-5
-4
-3
-2
-1
1
2
3
x
-8
-6
-4
-2
2
4
6
8
f (x)
c

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

-5
-4
-3
-2
-1
1
2
3
4
5
x
-8
-6
-4
-2
2
4
6
8
f (x)
d

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

-3
-2
-1
1
2
3
4
x
-18
-16
-14
-12
-10
-8
-6
-4
-2
g (x)
e

f \left( x \right)

-7
-6
-5
-4
-3
-2
-1
1
2
x
-7
-6
-5
-4
-3
-2
-1
1
2
3
f(x)
f

f \left( x \right)

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

Consider the function y = 4 x - 3.

a

Find the gradient function.

b

Sketch the graph of the gradient function.

5

For each of the following functions, sketch the gradient function:

a
f \left( x \right) = - 5 x + 8
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
f(x)
b

y = x^{2}

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
c

f \left( x \right) = \left(x + 1\right)^{2}

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f(x)
d

y = x^{2} - 1

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

f \left( x \right) = x \left(x - 4\right) \left(x + 2\right)

-8
-6
-4
-2
2
4
6
8
x
-18
-16
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
f (x)
6

Consider the function y = \left(x - 5\right)^{2} - 3 graphed below:

a

State the x-intercept of the gradient function.

b

For x \lt 5, state whether the values of the gradient function are above or below the x-axis.

c

For x \gt 5, state whether the values of the gradient function are above or below the x-axis.

-1
1
2
3
4
5
6
7
8
9
10
11
x
-4
-3
-2
-1
1
2
3
4
y
7

Consider the function y = - \left(x + 7\right)^{2} + 5 graphed below:

a

State the x-intercept of the gradient function.

b

For x \lt - 7, state whether the values of the gradient function are above or below the x-axis.

c

For x > - 7, state whether the values of the gradient function are above or below the x-axis.

-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
x
-4
-3
-2
-1
1
2
3
4
5
y
8

Consider the function y = - x^{2} - 10 x - 28 graphed below:

a

State the x-intercept of the gradient function.

b

For x \lt - 5, state whether the values of the gradient function are above or below the x-axis.

c

For x \gt - 5, state whether the values of the gradient function are above or below the x-axis.

-9
-8
-7
-6
-5
-4
-3
-2
-1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
y
9

Consider the function y = x^{2} + 10 x + 21 graphed below:

a

State the x-intercept of the gradient function.

b

For x \lt - 5, state whether the values of the gradient function are above or below the x-axis.

c

For x \gt - 5, state whether the values of the gradient function are above or below the x-axis.

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
-4
-3
-2
-1
1
2
3
4
y
10

Consider the function y = \left(x - 5\right)^{3} + 3:

a

State the coordinates of the point of inflection.

b

State the gradient of the curve at this point.

c

What feature does the gradient function have at x = 5?

d

For x \lt 5, state whether the values of the gradient function are above or below the x-axis.

e

For x \gt 5, state whether the values of the gradient function are above or below the x-axis.

f

What type of point is on the gradient function at x = 5?

1
2
3
4
5
6
7
8
9
x
1
2
3
4
5
6
y
11

Consider the function y = - \left(x + 7\right)^{3} - 3 :

a

State the coordinates of the point of inflection.

b

State the gradient of the curve at this point.

c

What feature does the gradient function have at x = - 7?

d

For x \lt - 7, state whether the values of the gradient function are above or below the x-axis.

e

For x \gt - 7, state whether the values of the gradient function are above or below the x-axis.

f

What type of point is on the gradient function at x = - 7?

-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
-6
-5
-4
-3
-2
-1
y
12

Consider the graph of the gradient function f' \left( x \right):

What can be said about the graph of f \left(x\right)?

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
f'(x)
13

For each of the following gradient functions, sketch a graph of a possible original function:

a
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f'(x)
b
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f'(x)
c
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f'(x)
d
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f'(x)
e
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
f'(x)
Types of stationary points
14

State the type of point that matches the following descriptions:

a

A point where the curve changes from decreasing to increasing.

b

A point where the curve changes from increasing to decreasing.

c

A point where the tangent is horizontal and the concavity changes about the point.

15

For each of the following functions:

i

Find the derivative.

ii

Find the coordinates of any stationary points.

iii

Classify each stationary point.

a
y = - 6 x^{2} + 84 x - 29
b
y = x^{3} - 21 x^{2} + 144 x - 19
c
f \left( x \right) = \left(x + 3\right)^{2} \left(x + 6\right)
d
f \left( x \right) = \left(x + 5\right)^{3} + 4
e
f \left( x \right) = - \dfrac{x^{3}}{3} + \dfrac{13 x^{2}}{2} - 30 x + 10
f
f \left( x \right) = \left( 4 x + 5\right) \left(x + 1\right)
g

f \left( x \right) = 134 - 300 x + 240 x^{2} - 64 x^{3}

h

f \left( x \right) = \left(x^{2} - 9\right)^{2} + 4

16

Consider the parabola with equation y = 5 + x - x^{2}.

a

Find the coordinates of the vertex of the parabola.

b

State the gradient of the tangent to the parabola at the vertex.

c

What type of stationary point is at the vertex of this parabola?

17

Consider the function f \left( x \right) = x^{2} + 4 x + 9.

a

Find an equation for the gradient function f' \left( x \right).

b

State the interval in which the function is increasing.

c

State the interval in which the function is decreasing.

d

Find the coordinates of the stationary point.

e

Classify the stationary point.

18

Consider the function f \left( x \right) = 4 x^{3} + 5 x^{2} - 4.

a

Find an equation for the gradient function f' \left( x \right).

b

Find the coordinates of the stationary points.

c

Complete the table of values:

x-2-\dfrac{5}{6}-\dfrac{1}{2}01
f'\left( x \right)00
d

Hence determine the:

i

Local minimum

ii

Local maximum

e

Is - 4 the absolute minimum value of the function? Explain your answer.

19

Consider the function f \left( x \right) = 3 x^{2} - 54 x + 241.

a

Find f' \left( x \right).

b

Find the x-coordinate of the stationary point.

c

Classify the stationary point.

Graphs of functions
20

Sketch the linear function for which f \left( 0 \right) = 1 and f' \left( 2 \right) = 3.

21

Sketch the quadratic function, f \left( x \right), that satisfies the following conditions:

a
  • f \left( 0 \right) = - 18
  • f \left( 3 \right) = 0
  • f \left( 6 \right) = 6
  • f' \left( 6 \right) = 0

  • f' \left( x \right) > 0 for x < 6

b
  • f \left( 0 \right) = 16
  • f \left( 2 \right) = 0
  • f \left( 5 \right) = - 9
  • f' \left( 5 \right) = 0

  • f' \left( x \right) < 0 for x < 5

c
  • f \left( 0 \right) = 5
  • f \left( - 2 \right) = 0
  • f' \left( 3 \right) = 0

  • f' \left( x \right) > 0 for x < 3

d

  • f \left( 0 \right) = 10

  • f \left( - 2 \right) = 0

  • f \left( - 6 \right) = - 8

  • f' \left( - 6 \right) = 0

  • f' \left( x \right) < 0 for x < - 6

22

Sketch the cubic function, f \left( x \right), that satisfies the following conditions:

a
  • f \left( 0 \right) = 7
  • f \left( - 2 \right) = 0
  • f \left( - 4 \right) = - 1
  • f' \left( - 4 \right) = 0

  • f' \left( x \right) > 0 for x < - 4

  • f' \left( x \right) > 0 for x > - 4

b
  • f' \left( 2 \right) = 0
  • f' \left( - 3 \right) = 0

  • f' \left( x \right) < 0 for - 3 < x < 2

  • f' \left( x \right) > 0 elsewhere

23

Sketch the quartic function, f \left( x \right), that satisfies the following conditions:

a
  • f' \left( - 1 \right) = 0
  • f' \left( 4 \right) = 0

  • f' \left( x \right) > 0 for x > 4

  • f' \left( x \right) < 0 elsewhere

b
  • f \left( 0 \right) = 0
  • f' \left( 0 \right) = 0
  • f' \left( 2 \right) = 0
  • f' \left( - 2 \right) = 0

  • f' \left( x \right) > 0 for x < - 2, 0 < x < 2

  • f' \left( x \right) < 0 elsewhere

24

Consider the equation of the parabola y = 3 x^{2} - 18 x + 24.

a

Find the x-intercepts.

b

Find the y-intercept.

c

Find \dfrac{dy}{dx}.

d

Find the stationary point.

e

Classify the stationary point.

f

Sketch the graph of the parabola.

25

For each of the following functions:

i

Find the y-intercept.

ii

Find the x-intercepts.

iii

Find f' \left( x \right).

iv

Hence find the x-coordinates of the stationary points.

v

Classify the stationary points.

vi

Sketch the graph of the function.

a
f \left( x \right) = 9 x^{2} + 18 x - 16
b
f \left( x \right) = \left( 4 x + 5\right)^{2} \left(x - 1\right)
c
f \left( x \right) = \left( 2 x - 1\right)^{2} \left(1 - x\right)
d
f \left( x \right) = \left(x + 2\right)^{3}-1
e
f \left( x \right) = x^{3} + 11 x^{2} + 24 x
f
f \left( x \right) = \left(x^{2} - 4\right)^{2} + 4
g
f(x)=2\left(x-1\right)^3 - 16
h
f(x)=x^4 - 4x^3
i
f(x) = x^3 + 5x^2
j
f(x)=x^2(x-3)^2
26

Sketch the graph of the following functions showing all stationary points:

a
y=2x^3-12x^2+18x-8
b
y=3x^5-20x^3 + 6
c
y=2x^3+3x^2-36x+5
d
y=20+4x^3-x^4
e
y=-3x^4+16x^3-24x^2+30
f
y=x^3-6x^2-15x+7
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