Problem solving — Transforming Functions

1Problem 1 of 36

Evaluate & solve. Let

f(x) = 2x + 3

.

(a) Find

f(5)

.
(b) Find

f(-1)

.
(c) Find the value of

x

for which

f(x) = 11

.

Working space

2Problem 2 of 36

Natural domains. State the largest natural domain of each function over

\mathbb{R}

.

(a)

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

(b)

g(x) = \sqrt{x - 1}

(c)

h(x) = \dfrac{1}{\sqrt{4 - x}}

Working space

3Problem 3 of 36

Vertical line test. State whether each relation defines

y

as a function of

x

. Justify.

(a)

y = x^2

(b)

x = y^2

(c)

x^2 + y^2 = 9

(d)

y = |x|

Working space

4Problem 4 of 36

Composition & inverse [EXT]. Let

f(x) = 2x + 5

and

g(x) = x^2 - 3

.

(a) Find

f(g(3))

.
(b) Find a simplified expression for

(f \circ g)(x)

.
(c) Find

f^{-1}(x)

and state its domain.

Working space

5Problem 5 of 36

Domain of a composition [EXT]. Let

f(x) = \sqrt{x}

and

g(x) = 5 - x^2

.

(a) Find

(f \circ g)(x)

.
(b) State its largest natural domain.
(c) State its range.

Working space

6Problem 6 of 36

Piecewise function. Let

f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & 0 \leq x \leq 3 \\ 9 & x > 3 \end{cases}

.

(a) Find

f(-2)

,

f(0)

,

f(2)

,

f(5)

.
(b) Sketch

f

for

-3 \leq x \leq 5

.
(c) State the range of

f

.

Working space

7Problem 7 of 36

Absolute value [EXT]. Solve and sketch.

(a) Solve

|2x - 3| = 7

.
(b) Solve

|x - 1| < 4

and write the answer in interval notation.
(c) Sketch

y = |x - 2| - 1

, marking

x

- and

y

-intercepts.

Working space

8Problem 8 of 36

Modelling with a linear function. A car rental charges a fixed daily fee

F

(CHF) plus a charge

c

(CHF) per km. A driver pays CHF 95 for 200 km in a day, and CHF 125 for 350 km in a day.

(a) Write a function

C(d) = cd + F

for the cost in terms of the distance

d

(km).
(b) Find

c

and

F

.
(c) Predict the cost of a 500 km day.

Working space

9Problem 9 of 36

Finding f from data — quadratic. A quadratic function

f

passes through

(0, 3)

,

(1, 6)

, and

(2, 13)

.

(a) Assume

f(x) = ax^2 + bx + c

. Set up three equations.
(b) Solve to find

a

,

b

,

c

.
(c) State

f(-1)

.

Working space

10Problem 10 of 36

Range of a quadratic. Let

f(x) = x^2 - 4x + 7

.

(a) Express

f(x)

in vertex form.
(b) State the minimum value and where it occurs.
(c) State the range of

f

.

Working space

11Problem 11 of 36

Inverse from a graph [EXT]. The graph of

f

is a straight line through

(-2, 1)

and

(4, 4)

.

(a) Find

f(x)

.
(b) Find

f^{-1}(x)

.
(c) On a single set of axes, sketch

y = f(x)

,

y = f^{-1}(x)

and

y = x

. State the symmetry.

Working space

12Problem 12 of 36

Mini-investigation — natural domain. Three students propose different formulas for the same function:

- Anya:

f(x) = \dfrac{x^2 - 1}{x - 1}

- Bao:

f(x) = x + 1

- Cara:

f(x) = x + 1

for

x \neq 1

(a) Compute

f(2)

using each formula.
(b) State the natural domain of Anya's formula.
(c) Are Anya's and Bao's functions equal? Justify.
(d) Whose formula is most precise? Justify.

Working space

13Problem 13 of 36

Factorising. Solve each quadratic by factorising.

(a)

x^2 - 7x + 12 = 0

(b)

2x^2 - 5x - 3 = 0

(c)

x^2 - 9 = 0

Working space

14Problem 14 of 36

Completing the square. Let

f(x) = x^2 + 8x + 10

.

(a) Express

f(x)

in vertex form.
(b) Hence solve

f(x) = 0

, giving exact answers in surd form.
(c) State the range of

f

.
(d) Describe the single transformation that maps

y = x^2

onto

y = (x + 5)^2

.

Working space

15Problem 15 of 36

Quadratic formula. Solve, giving exact answers.

(a)

x^2 - 4x + 1 = 0

(b)

2x^2 + 5x - 3 = 0

(c)

3x^2 - 7x + 2 = 0

Working space

16Problem 16 of 36

Graph features. For

f(x) = x^2 - 6x + 8

:

(a) Find the

x

- and

y

-intercepts.
(b) Find the vertex.
(c) State the axis of symmetry.
(d) Sketch the graph.

Working space

17Problem 17 of 36

Quadratic modelling — projectile. A ball is thrown vertically. Its height (m) above ground after

t

seconds is

h(t) = -5t^2 + 20t + 1.5

.

(a) Find the height at

t = 0

and explain what it represents.
(b) Find the time at which the ball reaches its maximum height.
(c) Find the maximum height.
(d) Find, to 3 s.f., the time at which the ball hits the ground.

Working space

18Problem 18 of 36

Quadratic modelling — fountain. A water jet follows

h(x) = -(x - 4)^2 + 9

(

h

height in m,

x

horizontal distance in m).

(a) Find the maximum height and where it occurs.
(b) Find

h(0)

and explain in context.
(c) Find the

x

-values at which the jet returns to ground level.

Working space

19Problem 19 of 36

Optimisation — area. A farmer uses 80 m of fencing to enclose a rectangular field with one side along a straight river (no fencing needed on that side).

(a) Let the width perpendicular to the river be

x

m. Express the length along the river and the area in terms of

x

.
(b) Find the value of

x

that maximises the area.
(c) State the maximum area and the corresponding length along the river.

Working space

20Problem 20 of 36

Discriminant — tangency [EXT]. The line

y = 2x + 1

meets the curve

y = x^2 + k

at exactly one point.

(a) Show that

x^2 - 2x + (k - 1) = 0

.
(b) Find

k

for which the line is tangent.
(c) Find the coordinates of the point of tangency.

Working space

21Problem 21 of 36

Inequality [EXT]. Solve

x^2 - x - 6 \geq 0

. Express your answer in interval notation, and on a number line.

Working space

22Problem 22 of 36

Quadratic from data. A quadratic

f(x) = ax^2 + bx + c

has

f(0) = 5

,

f(1) = 6

,

f(2) = 13

.

(a) Set up three equations in

a

,

b

,

c

.
(b) Solve for

a

,

b

,

c

.
(c) State the axis of symmetry and the vertex.

Working space

23Problem 23 of 36

Word problem — consecutive integers. The product of two consecutive positive integers is 156.

(a) Let the smaller be

n

. Write a quadratic equation in

n

.
(b) Solve to find the integers.

Working space

24Problem 24 of 36

Two-variable optimisation. A rectangular poster of total area 200 cm

^2

has a 2 cm margin on all four sides. The printed area inside the margins is to be maximised, subject to the poster's overall dimensions being integer multiples of 1 cm.

(a) Let the poster have width

w

cm and height

\frac{200}{w}

cm. Write the printed area

A

in terms of

w

.
(b) Sketch (or describe)

A(w)

for

w > 4

and find the value of

w

that maximises

A

.
(c) State the maximum printed area.

Working space

25Problem 25 of 36

Single transformations. Describe the single transformation that maps the first graph onto the second.

(a)

y = x^2 \to y = (x + 5)^2

(b)

y = x^2 \to y = x^2 + 3

(c)

y = x^2 \to y = -x^2

(d)

y = x^2 \to y = 4x^2

Working space

26Problem 26 of 36

Pinpoint a transformed feature. The graph of

y = f(x)

has a minimum at

(-1, 4)

and

y

-intercept

(0, 7)

.

(a) State the new minimum and

y

-intercept of

y = f(x) - 5

.
(b) State the new minimum and

y

-intercept of

y = f(x + 3)

.
(c) State the new turning point and

y

-intercept of

y = -f(x)

. Is it now a min or max?
(d) State the new minimum and

y

-intercept of

y = f(x/2)

.

Working space

27Problem 27 of 36

Translation chain. Starting from

y = x^2

:

(a) Translate 2 units right, then 3 units down. Write the equation.
(b) Reflect in the

x

-axis, then translate 1 unit left and 4 units up.
(c) Vertical stretch by 2, then horizontal translation 3 right.

Working space

28Problem 28 of 36

Vertex form to features. Let

f(x) = -2(x + 3)^2 + 5

.

(a) State the vertex and whether it is a max or min.
(b) State the axis of symmetry.
(c) State the

y

-intercept.
(d) Describe how

y = f(x)

is obtained from

y = x^2

as a sequence of transformations.

Working space

29Problem 29 of 36

Inverse chain. Starting from

y = x^2

, the transformations applied are: translate 4 right, vertical stretch by 3, then reflect in the

x

-axis.

(a) Write the final equation.
(b) State a sequence of transformations that returns the final graph to

y = x^2

.

Working space

30Problem 30 of 36

Find a transformation from data. The graph of

y = f(x)

passes through

(0, 1)

,

(1, 4)

,

(2, 13)

. After the transformation

g(x) = f(x - 2) + 3

, the graph passes through which corresponding points?

(a) State the three points on the graph of

y = g(x)

.
(b) State

g(3)

.
(c) State

g(4)

.

Working space

31Problem 31 of 36

Stretching. The graph of

y = f(x)

has

x

-intercepts at

x = 1

and

x = 5

, and

y

-intercept

(0, -5)

.

(a) State the corresponding intercepts of

y = 2f(x)

.
(b) State the corresponding intercepts of

y = f(2x)

.
(c) State the corresponding intercepts of

y = -f(x)

.

Working space

32Problem 32 of 36

Sequence on a sinusoid. Starting from

y = \sin x

:

(a) Write the equation of the graph after a vertical stretch by 3 and a horizontal compression by factor 2.
(b) State its amplitude and period.
(c) Apply the further transformation: translation 1 unit up. Write the new equation.

Working space

33Problem 33 of 36

Match the equation. The graph of

y = f(x)

has key features at

(0, 3)

,

(2, 0)

and

(4, -3)

. State the equation for the curve that passes through

(0, 6)

,

(2, 0)

,

(4, -6)

in terms of

f

.

Working space

34Problem 34 of 36

Asymptotes and intercepts. The graph of

y = \dfrac{1}{x}

has vertical asymptote

x = 0

and horizontal asymptote

y = 0

.

(a) State the asymptotes of

y = \dfrac{1}{x - 3} + 2

.
(b) Describe the transformations from

y = \dfrac{1}{x}

to that graph.
(c) Find the

y

-intercept of

y = \dfrac{1}{x - 3} + 2

.

Working space

35Problem 35 of 36

Composing transformations. Let

f(x) = x^2

and

g(x) = -3 f(x - 1) + 4

.

(a) Describe the chain of transformations.
(b) State the vertex of

y = g(x)

and whether it is a max or min.
(c) Find the

y

-intercept of

y = g(x)

.

Working space

36Problem 36 of 36

Investigation — order of operations. Consider the two chains starting from

y = f(x)

:

- Chain 1: vertical stretch by 3, then translate 2 up.
- Chain 2: translate 2 up, then vertical stretch by 3.

(a) Write the equation of the graph for each chain.
(b) Are they the same? If not, explain why the order matters.
(c) Find a different pair of operations whose order does not matter and justify.

Working space

Problem-solving Pack

Problem-solving Pack