Dossier A — Fluidité — 11.5 Transformations of Functions

BronzeQuestions 1–10

1.
Describe the single transformation that maps $y = x^2$ to $y = (x + 5)^2$ .
2.
Describe the single transformation that maps $y = f(x)$ to $y = f(x) + 4$ .
3.
Describe the transformation mapping $y = f(x)$ to $y = -f(x)$ .
4.
Describe the transformation mapping $y = f(x)$ to $y = 3 \cdot f(x)$ .
5.
The graph of $y = f(x)$ has $y$ -intercept $(0, 4)$ . State the $y$ -intercept of $y = f(x) - 3$ .
6.
The graph of $y = f(x)$ has minimum at $(2, -3)$ . State the new minimum of $y = f(x - 4)$ .
7.
The graph of $y = f(x)$ contains the point $(2, 5)$ . What point does the graph of $y = f(-x)$ contain?
8.
State the vertex of $y = (x - 2)^2 + 5$ .
9.
Describe the transformation mapping $y = f(x)$ to $y = f(2x)$ .
10.
Describe the single transformation from $y = \sqrt{x}$ to $y = \sqrt{x} + 3$ .

SilverQuestions 11–20

11.
The graph of $y = x^2$ is translated 3 units right and 2 units up. Write the equation of the new graph.
12.
The graph of $y = x^2$ is reflected in the $x$ -axis and translated 4 units up. Write the new equation.
13.
Describe the single transformation mapping $y = x^2$ to $y = 3x^2$ .
14.
The graph of $y = f(x)$ has minimum at $(-1, 4)$ . State the coordinates of the new minimum for $y = f(x) - 5$ .
15.
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.
16.
The graph of $y = f(x)$ contains the points $(0, 2)$ and $(4, -1)$ . State the points after the transformation $y = 2f(x)$ .
17.
Starting from $y = x^2$ , translate 2 left, then reflect in the $x$ -axis. Write the final equation.
18.
The graph of $y = f(x)$ contains $(6, 8)$ . State the point on the graph of $y = f(2x)$ corresponding to this.
19.
The graph of $y = f(x)$ has a maximum at $(2, 5)$ . State the maximum of $y = f(x - 1) + 3$ .
20.
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 21–30

21.
Starting from $y = x^2$ , reflect in the $x$ -axis, then translate 2 right and 4 up. State the final equation and the vertex.
22.
Describe a sequence of transformations from $y = x^2$ to $y = 2(x - 3)^2 - 1$ .
23.
If $f(x) = x^2$ , and $g(x) = f(x - 1) + 4$ , find the $x$ -values for which $g(x) = 8$ .
24.
The point $(2, 3)$ is on the graph of $y = f(x)$ . State the corresponding point on the graph of $y = -f(x) + 4$ .
25.
The graph of $y = f(x)$ has key features at $x = 1$ and $y$ -value 3. State the corresponding features on $y = 2f(3x)$ .
26.
A parabola has vertex $(2, -1)$ and passes through $(0, 7)$ . Find its equation in vertex form.
27.
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 $y = -f(x)$ .
28.
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 $f$ .
29.
Express $y = 2x^2 + 12x + 13$ in vertex form, then describe the chain of transformations from $y = x^2$ .
30.
The graph of $y = f(x)$ has its maximum at $(0, 4)$ . Where is the maximum of $y = f(2(x - 3))$ ?

PlatinumQuestions 31–40

31.
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.
32.
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 $f$ ).
33.
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.
34.
A function $g(x) = (x + 3)^2 - 4$ is obtained from $y = x^2$ by a sequence of transformations. State the sequence (in order).
35.
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.
36.
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.
37.
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.
38.
The function $y = x^2$ is translated so that its new vertex is $(4, -7)$ . Find $h$ and $k$ if the new equation is $y = (x - h)^2 + k$ .
39.
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.
40.
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?

Pack A — Fluency