Pack A — Fluency

Mathematics

Year 11 · Transforming Functions

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–30

1.
Let $f(x) = 2x + 3$ . Find $f(5)$ .
2.
For $f(x) = 2x + 3$ , find $x$ such that $f(x) = 11$ .
3.
State the largest natural domain over $\mathbb{R}$ of $f(x) = \dfrac{1}{x - {a}}$ .
4.
State the largest natural domain over $\mathbb{R}$ of $g(x) = \sqrt{x - {a}}$ .
5.
Let $f(x) = x^2 + -3x + 2$ . Find $f(4)$ .
6.
A vertical line drawn through the graph of $y = x^2$ meets the curve in at most one point. Is $y = x^2$ a function of $x$ ?
7.
State the range of $f(x) = 2x + 1$ defined on $\mathbb{R}$ .
8.
For $f(x) = 3x - 2$ , find the image of $x = -2$ .
9.
State the largest natural domain of $h(x) = \dfrac{1}{\sqrt{x - 4}}$ .
10.
The function $f$ is defined by the table: $f(1) = 4$ , $f(2) = 7$ , $f(3) = 10$ . Find a formula for $f(n)$ if it is linear.
11.
Solve $x^2 - 7x + 12 = 0$ by factorising.
12.
Solve $x^2 - 16 = 0$ .
13.
Solve $x^2 + 5x + 6 = 0$ using factorising.
14.
State the coordinates of the vertex of $y = (x - 3)^2 + -4$ and whether it is a min or max.
15.
State the $y$ -intercept of $y = x^2 - 5x + 7$ .
16.
Solve $2x^2 - 5x - 3 = 0$ by factorising.
17.
State the axis of symmetry of $y = x^2 - 6x + 8$ .
18.
Solve $x^2 - 4x - 5 = 0$ using the quadratic formula.
19.
Does $y = -2x^2 + 3x + 1$ open upwards or downwards?
20.
State the $x$ -intercepts of $y = (x - 2)(x + 5)$ .
21.
Describe the single transformation that maps $y = x^2$ to $y = (x + 5)^2$ .
22.
Describe the single transformation that maps $y = f(x)$ to $y = f(x) + 4$ .
23.
Describe the transformation mapping $y = f(x)$ to $y = -f(x)$ .
24.
Describe the transformation mapping $y = f(x)$ to $y = 3 \cdot f(x)$ .
25.
The graph of $y = f(x)$ has $y$ -intercept $(0, 4)$ . State the $y$ -intercept of $y = f(x) - 3$ .
26.
The graph of $y = f(x)$ has minimum at $(2, -3)$ . State the new minimum of $y = f(x - 4)$ .
27.
The graph of $y = f(x)$ contains the point $(2, 5)$ . What point does the graph of $y = f(-x)$ contain?
28.
State the vertex of $y = (x - 2)^2 + 5$ .
29.
Describe the transformation mapping $y = f(x)$ to $y = f(2x)$ .
30.
Describe the single transformation from $y = \sqrt{x}$ to $y = \sqrt{x} + 3$ .

SilverQuestions 31–60

31.
Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$ . Find $f(g( 3))$ .
32.
Let $f(x) = 2x + 5$ and $g(x) = x^2 - 3$ . Find a simplified expression for $(f \circ g)(x)$ .
33.
Find $f^{-1}(x)$ for $f(x) = 2x + 5$ .
34.
State the domain and range of $f(x) = \dfrac{1}{x - 2}$ .
35.
For $f(x) = 2x + 5$ and $g(x) = -x + 8$ , solve $f(x) = g(x)$ .
36.
State the domain and range of $f(x) = |x - 3|$ .
37.
For what values of $x$ is $f(x) = \sqrt{2x - 6}$ defined?
38.
For $f(x) = x^2$ and $g(x) = x + 6$ , find all $x$ with $f(x) = g(x)$ .
39.
If $f(3) = 7$ and $f$ is one-to-one, state $f^{-1}(7)$ .
40.
Let $f(x) = \begin{cases} 2x + 1 & x < 0 \\ x^2 & x \geq 0 \end{cases}$ . Find $f(-3)$ and $f(2)$ .
41.
Express $x^2 + 8x + 10$ in the form $(x + p)^2 + q$ .
42.
Find the vertex of $y = x^2 - 6x + 11$ .
43.
Solve $2x^2 - x - 6 = 0$ by factorising.
44.
Expand $(x - 3)(x + 5)$ and write in $ax^2 + bx + c$ form.
45.
Write $y = 2(x - 3)^2 - 5$ in expanded form $y = ax^2 + bx + c$ .
46.
Solve $x^2 + 8x + 10 = 0$ exactly, leaving any irrational answer in surd form.
47.
A number plus its square is 30. Find the number.
48.
For $y = (x - 3)^2 - 4$ , state (a) the vertex, (b) the $x$ -intercepts.
49.
Express $y = 2x^2 - 8x + 5$ in vertex form.
50.
A rectangle has length $(x + 3)$ cm and width $x$ cm. Its area is 18 cm $^2$ . Find $x$ .
51.
The graph of $y = x^2$ is translated 3 units right and 2 units up. Write the equation of the new graph.
52.
The graph of $y = x^2$ is reflected in the $x$ -axis and translated 4 units up. Write the new equation.
53.
Describe the single transformation mapping $y = x^2$ to $y = 3x^2$ .
54.
The graph of $y = f(x)$ has minimum at $(-1, 4)$ . State the coordinates of the new minimum for $y = f(x) - 5$ .
55.
The graph of $y = f(x)$ has minimum at $(-1, 4)$ and $y$ -intercept $(0, 7)$ . State the new turning point and $y$ -intercept of $y = -f(x)$ , and say whether it becomes a maximum or minimum.
56.
The graph of $y = f(x)$ contains the points $(0, 2)$ and $(4, -1)$ . State the points after the transformation $y = 2f(x)$ .
57.
Starting from $y = x^2$ , translate 2 left, then reflect in the $x$ -axis. Write the final equation.
58.
The graph of $y = f(x)$ contains $(6, 8)$ . State the point on the graph of $y = f(2x)$ corresponding to this.
59.
The graph of $y = f(x)$ has a maximum at $(2, 5)$ . State the maximum of $y = f(x - 1) + 3$ .
60.
The graph of $y = \frac{1}{x}$ has a vertical asymptote at $x = 0$ . State the vertical asymptote of $y = \frac{1}{x - 3}$ .

GoldQuestions 61–90

61.
Let $f(x) = 2x + 5$ . Find $f^{-1}(x)$ and state its domain.
62.
Let $f(x) = \sqrt{x}$ and $g(x) = x - 4$ . State $(f \circ g)(x)$ and its largest natural domain.
63.
State the range of $f(x) = |x - 3| + 2$ .
64.
For $f(x) = \lfloor x \rfloor$ (the floor function), find $f(2.7)$ and $f(-1.4)$ .
65.
$f(x) = x^2$ is not invertible on $\mathbb{R}$ . State a domain restriction that makes it invertible, and give the inverse.
66.
A graph passes through $(0, 4)$ and decreases monotonically, approaching $y = 0$ but never reaching it. Is this consistent with $f(x) = 4 \cdot (0.5)^x$ ? Justify.
67.
Let $f(x) = 2x + 1$ and $g(x) = \dfrac{1}{x}$ for $x \neq 0$ . Find $(g \circ f)(x)$ and state its domain.
68.
For $f(x) = |2x - 4|$ , solve $f(x) = 6$ .
69.
The cost (CHF) of producing $n$ bottles is $C(n) = 0.5n + 200$ . State (a) the meaning of the gradient, (b) the meaning of $C(0)$ .
70.
The graph of $y = f^{-1}(x)$ is the reflection of the graph of $y = f(x)$ in which line?
71.
Express $f(x) = x^2 + 8x + 10$ in vertex form, then state the range of $f$ .
72.
A quadratic $y = x^2 + bx + 7$ has its minimum value at $x = 3$ . Find $b$ .
73.
How many real roots does $2x^2 - 4x + 3 = 0$ have? Use the discriminant.
74.
The line $y = 2x + 1$ is tangent to the curve $y = x^2 + k$ . Find $k$ .
75.
Solve $x^2 - 5x + 6 \leq 0$ .
76.
A monic quadratic has roots $-2$ and $5$ . Write it in expanded form.
77.
A ball thrown follows $h(t) = -5t^2 + 20t + 1$ ( $h$ in m, $t$ in s). Find (a) the maximum height, (b) the time it occurs.
78.
For $x^2 + 5x - 6 = 0$ , find the sum and product of the roots without solving.
79.
A fountain jet has $h(x) = -(x - 4)^2 + 9$ ( $h$ height in m, $x$ horizontal distance in m). Find (a) max height, (b) range of $x$ above ground.
80.
For what range of $k$ does $x^2 + kx + 9 = 0$ have no real roots?
81.
Starting from $y = x^2$ , reflect in the $x$ -axis, then translate 2 right and 4 up. State the final equation and the vertex.
82.
Describe a sequence of transformations from $y = x^2$ to $y = 2(x - 3)^2 - 1$ .
83.
If $f(x) = x^2$ , and $g(x) = f(x - 1) + 4$ , find the $x$ -values for which $g(x) = 8$ .
84.
The point $(2, 3)$ is on the graph of $y = f(x)$ . State the corresponding point on the graph of $y = -f(x) + 4$ .
85.
The graph of $y = f(x)$ has key features at $x = 1$ and $y$ -value 3. State the corresponding features on $y = 2f(3x)$ .
86.
A parabola has vertex $(2, -1)$ and passes through $(0, 7)$ . Find its equation in vertex form.
87.
The graph of $y = f(x)$ has $x$ -intercepts at $x = 1$ and $x = 5$ , and $y$ -intercept $(0, -5)$ . State the corresponding intercepts of $y = -f(x)$ .
88.
The graph of $y = g(x)$ is obtained from $y = f(x)$ by a reflection in the $y$ -axis followed by a translation 2 units down. Write $g(x)$ in terms of $f$ .
89.
Express $y = 2x^2 + 12x + 13$ in vertex form, then describe the chain of transformations from $y = x^2$ .
90.
The graph of $y = f(x)$ has its maximum at $(0, 4)$ . Where is the maximum of $y = f(2(x - 3))$ ?

PlatinumQuestions 91–120

91.
Let $f(x) = 2x + 1$ and $g(x) = \dfrac{x}{x - 1}$ for $x \neq 1$ . Find $(g \circ f)(x)$ and state its domain.
92.
For $f(x) = \dfrac{1}{x - 1}$ , find $(f \circ f)(x)$ and identify the values that are fixed by $f \circ f$ .
93.
Find the inverse of $f(x) = \dfrac{2x + 3}{x - 1}$ for $x \neq 1$ , and state its domain.
94.
State the largest natural domain of $f(x) = \dfrac{1}{\sqrt{x^2 - 4}}$ .
95.
For $f(x) = \begin{cases} 2x + a & x < 1 \\ x^2 & x \geq 1 \end{cases}$ , find $a$ such that $f$ is continuous at $x = 1$ .
96.
Find the range of $f(x) = x^2 - 4x + 7$ on $\mathbb{R}$ .
97.
The graph of $y = f(x)$ passes through $(1, 4)$ and $(3, 10)$ . If $f$ is linear, find $f^{-1}(x)$ .
98.
Let $f(x) = 2x - 1$ . Solve $f(f(x)) = 11$ .
99.
Let $f(x) = (x - 2)^2$ for $x \geq 2$ . Find $f^{-1}(x)$ and its domain.
100.
A circle of radius 5 centred at the origin has equation $x^2 + y^2 = 25$ . Explain why this is **not** a function of $x$ , and write the two functions $y = f_1(x)$ and $y = f_2(x)$ that together describe the circle.
101.
A rectangle has perimeter 26 cm and area 40 cm $^2$ . Find its dimensions.
102.
For what values of $k$ does the line $y = x + k$ intersect $y = x^2$ at two distinct points?
103.
A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$ , $(1, 8)$ and $(3, 2)$ . Find $a$ , $b$ , $c$ .
104.
Express $y = x^2 - 6x + 11$ in vertex form. Hence describe the transformation from $y = x^2$ to $y = x^2 - 6x + 11$ .
105.
A parabola has vertex $(2, 3)$ and passes through $(5, 30)$ . Find its equation.
106.
Find the points of intersection of $y = x^2 + 2x - 3$ and $y = x + 1$ .
107.
Find the minimum value of $f(x) = 3x^2 - 12x + 7$ .
108.
Show that for all real $k$ , the equation $x^2 + 2kx + (k^2 + 1) = 0$ has no real solutions.
109.
A farmer has 40 m of fencing to enclose a rectangular pen against a wall (so one side is the wall). Find the dimensions giving the maximum area.
110.
A ball is thrown and its height (m) at times $t = 0, 1, 2$ is recorded as $1.5, 5, 7.5$ m. Assuming $h(t) = at^2 + bt + c$ , find $a$ , $b$ , $c$ and predict $h(3)$ .
111.
The graph of $y = f(x)$ has minimum at $(2, -3)$ . Find the new minimum after: shift right 3, then reflect in the $x$ -axis, then shift up 5.
112.
The graph of $y = g(x)$ is obtained from $y = f(x)$ by translating 3 right then stretching vertically by factor 2. Write the inverse transformation chain (from $g$ back to $f$ ).
113.
A bounded region between $y = f(x)$ and the $x$ -axis has area 12. State the area of the region between $y = 3 f(x/2)$ and the $x$ -axis.
114.
A function $g(x) = (x + 3)^2 - 4$ is obtained from $y = x^2$ by a sequence of transformations. State the sequence (in order).
115.
The graph of $y = \sin x$ is transformed so that its amplitude becomes 3, its period is $\pi$ , and it is shifted up by 2. Write the equation.
116.
Let $f(x) = (x - 2)^2$ and $g(x) = f(x + 4) - 1$ . Find the vertex of $y = g(x)$ and write $g$ in expanded form.
117.
The graph of $y = f(x)$ passes through $(1, 2)$ and $(3, 8)$ . State the corresponding points on $y = f(x - 2) + 5$ , and find the average rate of change of the new function between them.
118.
The function $y = x^2$ is translated so that its new vertex is $(4, -7)$ . Find $h$ and $k$ if the new equation is $y = (x - h)^2 + k$ .
119.
The graph of $y = f(x)$ passes through $(2, 5)$ . State the corresponding point on the graph of $y = f^{-1}(x)$ , and describe the geometric relationship.
120.
Is the function $f(x) = x^3 - 4x$ even, odd, or neither? After a translation 2 units right, is the resulting function even, odd, or neither?