Corrigé
11.5 Transformations of Functions
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Describe the single transformation that maps $y = x^2$ to $y = (x + 5)^2$. | Translation 5 units left |
| 2 | Describe the single transformation that maps $y = f(x)$ to $y = f(x) + 4$. | Translation 4 units up |
| 3 | Describe the transformation mapping $y = f(x)$ to $y = -f(x)$. | Reflection in the $x$-axis |
| 4 | Describe the transformation mapping $y = f(x)$ to $y = 3 \cdot f(x)$. | Vertical stretch (dilation) by factor 3 |
| 5 | The graph of $y = f(x)$ has $y$-intercept $(0, 4)$. State the $y$-intercept of $y = f(x) - 3$. | $(0, 1)$ |
| 6 | The graph of $y = f(x)$ has minimum at $(2, -3)$. State the new minimum of $y = f(x - 4)$. | $(6, -3)$ |
| 7 | The graph of $y = f(x)$ contains the point $(2, 5)$. What point does the graph of $y = f(-x)$ contain? | $(-2, 5)$ |
| 8 | State the vertex of $y = (x - 2)^2 + 5$. | $(2, 5)$ |
| 9 | Describe the transformation mapping $y = f(x)$ to $y = f(2x)$. | Horizontal compression by factor $\dfrac{1}{2}$ |
| 10 | Describe the single transformation from $y = \sqrt{x}$ to $y = \sqrt{x} + 3$. | Translation 3 units up |
| 11 | The graph of $y = x^2$ is translated 3 units right and 2 units up. Write the equation of the new graph. | $y = (x - 3)^2 + 2$ |
| 12 | The graph of $y = x^2$ is reflected in the $x$-axis and translated 4 units up. Write the new equation. | $y = -x^2 + 4$ |
| 13 | Describe the single transformation mapping $y = x^2$ to $y = 3x^2$. | Vertical stretch by factor 3 |
| 14 | The graph of $y = f(x)$ has minimum at $(-1, 4)$. State the coordinates of the new minimum for $y = f(x) - 5$. | $(-1, -1)$ |
| 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. | Turning point $(-1, -4)$ — maximum; $y$-intercept $(0, -7)$ |
| 16 | The graph of $y = f(x)$ contains the points $(0, 2)$ and $(4, -1)$. State the points after the transformation $y = 2f(x)$. | $(0, 4)$ and $(4, -2)$ |
| 17 | Starting from $y = x^2$, translate 2 left, then reflect in the $x$-axis. Write the final equation. | $y = -(x + 2)^2$ |
| 18 | The graph of $y = f(x)$ contains $(6, 8)$. State the point on the graph of $y = f(2x)$ corresponding to this. | $(3, 8)$ |
| 19 | The graph of $y = f(x)$ has a maximum at $(2, 5)$. State the maximum of $y = f(x - 1) + 3$. | Max at $(3, 8)$ |
| 20 | The graph of $y = \frac{1}{x}$ has a vertical asymptote at $x = 0$. State the vertical asymptote of $y = \frac{1}{x - 3}$. | $x = 3$ |
| 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. | $y = -(x - 2)^2 + 4$; vertex $(2, 4)$ |
| 22 | Describe a sequence of transformations from $y = x^2$ to $y = 2(x - 3)^2 - 1$. | Vertical stretch ×2, then translate 3 right and 1 down. |
| 23 | If $f(x) = x^2$, and $g(x) = f(x - 1) + 4$, find the $x$-values for which $g(x) = 8$. | $x = -1$ or $x = 3$ |
| 24 | The point $(2, 3)$ is on the graph of $y = f(x)$. State the corresponding point on the graph of $y = -f(x) + 4$. | $(2, 1)$ |
| 25 | The graph of $y = f(x)$ has key features at $x = 1$ and $y$-value 3. State the corresponding features on $y = 2f(3x)$. | Feature at $x = \dfrac{1}{3}$ with $y$-value 6. |
| 26 | A parabola has vertex $(2, -1)$ and passes through $(0, 7)$. Find its equation in vertex form. | $y = 2(x - 2)^2 - 1$ |
| 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 $y = -f(x)$. | $x$-intercepts at $x = 1$ and $x = 5$ (unchanged); $y$-intercept $(0, 5)$. |
| 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 $f$. | $g(x) = f(-x) - 2$ |
| 29 | Express $y = 2x^2 + 12x + 13$ in vertex form, then describe the chain of transformations from $y = x^2$. | $y = 2(x + 3)^2 - 5$; vertical stretch ×2, translate 3 left, then 5 down. |
| 30 | The graph of $y = f(x)$ has its maximum at $(0, 4)$. Where is the maximum of $y = f(2(x - 3))$? | $(3, 4)$ |
| 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. | Maximum at $(5, 8)$ |
| 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 $f$). | Compress vertically by factor $\dfrac{1}{2}$, then translate 3 left. |
| 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. | 72 |
| 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). | Translate 3 left, then 4 down. |
| 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. | $y = 3\sin(2x) + 2$ |
| 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. | Vertex $(-2, -1)$; $g(x) = x^2 + 4x + 3$ |
| 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. | $(3, 7)$ and $(5, 13)$; average rate of change 3 (unchanged from $f$). |
| 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 $y = (x - h)^2 + k$. | $h = 4$, $k = -7$ |
| 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. | Corresponds to $(5, 2)$; reflection of $f$ in the line $y = x$. |
| 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? | $f$ is odd. After translation, $g(x) = (x - 2)^3 - 4(x - 2)$ is neither. |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Describe the single transformation that maps $y = x^2$ to $y = (x + -3)^2$. | Translation 3 units right |
| 2 | Describe the single transformation that maps $y = f(x)$ to $y = f(x) + -5$. | Translation 5 units down |
| 3 | Describe the transformation mapping $y = f(x)$ to $y = -f(x)$. | Reflection in the $y$-axis |
| 4 | Describe the transformation mapping $y = f(x)$ to $y = \frac{1}{2} \cdot f(x)$. | Vertical dilation by factor $\dfrac{1}{2}$ (compression) |
| 5 | The graph of $y = f(x)$ has $y$-intercept $(0, 4)$. State the $y$-intercept of $y = f(x) - -2$. | $(0, 6)$ |
| 6 | The graph of $y = f(x)$ has minimum at $(2, -3)$. State the new minimum of $y = f(x - -1)$. | $(1, -3)$ |
| 7 | The graph of $y = f(x)$ contains the point $(2, 5)$. What point does the graph of $y = f(-x)$ contain? | $(-3, 7)$ |
| 8 | State the vertex of $y = (x - 2)^2 + 5$. | $(-1, -4)$ |
| 9 | Describe the transformation mapping $y = f(x)$ to $y = f(\frac{1}{3}x)$. | Horizontal stretch by factor 3 |
| 10 | Describe the single transformation from $y = \sqrt{x}$ to $y = \sqrt{x} + 3$. | Translation 4 units right |
| 11 | The graph of $y = x^2$ is translated 3 units right and 2 units up. Write the equation of the new graph. | $y = (x + 4)^2 - 1$ |
| 12 | The graph of $y = x^2$ is reflected in the $x$-axis and translated 4 units up. Write the new equation. | $y = -(x + 3)^2$ |
| 13 | Describe the single transformation mapping $y = x^2$ to $y = 3x^2$. | Vertical compression by factor $\frac{1}{2}$ |
| 14 | The graph of $y = f(x)$ has minimum at $(-1, 4)$. State the coordinates of the new minimum for $y = f(x) - 5$. | $(-1, 7)$ |
| 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. | Turning point $(-2, 4)$ — still a minimum; $y$-intercept $(0, 7)$ |
| 16 | The graph of $y = f(x)$ contains the points $(0, 2)$ and $(4, -1)$. State the points after the transformation $y = 2f(x)$. | $(0, 5)$ and $(4, 2)$ |
| 17 | Starting from $y = x^2$, translate 2 left, then reflect in the $x$-axis. Write the final equation. | $y = -(x - 3)^2$ |
| 18 | The graph of $y = f(x)$ contains $(6, 8)$. State the point on the graph of $y = f(2x)$ corresponding to this. | $(18, 8)$ |
| 19 | The graph of $y = f(x)$ has a maximum at $(2, 5)$. State the maximum of $y = f(x - 1) + 3$. | Now a minimum at $(2, -3)$ |
| 20 | The graph of $y = \frac{1}{x}$ has a vertical asymptote at $x = 0$. State the vertical asymptote of $y = \frac{1}{x - 3}$. | $x = -2$ |
| 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. | $y = -[(x + 1)^2 - 3] = -(x + 1)^2 + 3$; vertex $(-1, 3)$ |
| 22 | Describe a sequence of transformations from $y = x^2$ to $y = 2(x - 3)^2 - 1$. | Reflect in $x$-axis, then translate 2 left and 5 up. |
| 23 | If $f(x) = x^2$, and $g(x) = f(x - 1) + 4$, find the $x$-values for which $g(x) = 8$. | $x = -5$ or $x = 1$ |
| 24 | The point $(2, 3)$ is on the graph of $y = f(x)$. State the corresponding point on the graph of $y = -f(x) + 4$. | $(2, 3)$ |
| 25 | The graph of $y = f(x)$ has key features at $x = 1$ and $y$-value 3. State the corresponding features on $y = 2f(3x)$. | Feature at $x = 2$ with $y$-value $\dfrac{3}{2}$. |
| 26 | A parabola has vertex $(2, -1)$ and passes through $(0, 7)$. Find its equation in vertex form. | $y = -2(x + 1)^2 + 3$ |
| 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 $y = -f(x)$. | $x$-intercepts at $x = -1$ and $x = -5$; $y$-intercept $(0, -5)$ unchanged. |
| 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 $f$. | $g(x) = -f(x - 3)$ |
| 29 | Express $y = 2x^2 + 12x + 13$ in vertex form, then describe the chain of transformations from $y = x^2$. | $y = -(x - 2)^2 + 3$; reflect in $x$-axis, translate 2 right, then 3 up. |
| 30 | The graph of $y = f(x)$ has its maximum at $(0, 4)$. Where is the maximum of $y = f(2(x - 3))$? | Maximum at $(-1, 10)$ |
| 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. | Minimum at $(3, -2)$ |
| 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 $f$). | Translate down 4, then reflect in the $x$-axis. |
| 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. | 12 |
| 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). | Vertical stretch ×2, reflect in $x$-axis, translate 1 right, then 5 up. |
| 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. | $y = \dfrac{1}{2}\sin\!\left(\dfrac{x}{2}\right) - 1$ |
| 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. | Vertex $(3, 3)$ — now a maximum; $g(x) = -(x - 3)^2 + 3$ |
| 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. | $(1, 4)$ and $(3, 16)$; average rate of change 6 (doubled). |
| 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 $y = (x - h)^2 + k$. | $h = -3$, $k = 2$ |
| 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. | Corresponds to $(3, -1)$; same reflection. |
| 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? | $f$ is even. $g(x) = (x - 3)^2 + 1$ is neither (translation in $x$ breaks $y$-axis symmetry). |
Problèmes — Solutions détaillées
**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) $y = x^2 \to y = 4x^2$
(a) Translation 5 units left. (b) Translation 3 units up. (c) Reflection in the $x$-axis. (d) Vertical stretch by factor 4.
**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 $y = f(x/2)$.
(a) $(-1, -1)$, $(0, 2)$. (b) $(-4, 4)$, $y$-intercept depends on $f(3)$. (c) $(0, -7)$, max at $(-1, -4)$. (d) $(-2, 4)$, $(0, 7)$ unchanged.
**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.
(a) $y = (x - 2)^2 - 3$. (b) $y = -(x + 1)^2 + 4$. (c) $y = 2(x - 3)^2$.
**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.
(a) Vertex $(-3, 5)$; maximum. (b) $x = -3$. (c) $(0, -13)$. (d) Vertical stretch ×2, reflect in $x$-axis, translate 3 left and 5 up.
**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 $y = x^2$.
(a) $y = -3(x - 4)^2$. (b) Reflect in the $x$-axis, vertically compress by factor $\frac{1}{3}$, then translate 4 left.
**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 $g(4)$.
(a) $(2, 4)$, $(3, 7)$, $(4, 16)$. (b) $g(3) = f(1) + 3 = 7$. (c) $g(4) = f(2) + 3 = 16$.
**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 $y = -f(x)$.
(a) $x = 1, 5$ unchanged; $y$-intercept $(0, -10)$. (b) $x = \frac{1}{2}, \frac{5}{2}$; $y$-intercept $(0, -5)$ unchanged. (c) $x = 1, 5$ unchanged; $y$-intercept $(0, 5)$.
**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.
(a) $y = 3\sin(2x)$. (b) Amplitude 3, period $\pi$. (c) $y = 3\sin(2x) + 1$.
**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 $f$.
$y = 2f(x)$.
**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 $y = \dfrac{1}{x - 3} + 2$.
(a) $x = 3$ and $y = 2$. (b) Translate 3 right and 2 up. (c) $(0, \frac{5}{3})$.
**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 $y = g(x)$.
(a) Translate 1 right, vertical stretch ×3, reflect in $x$-axis, translate 4 up. (b) $(1, 4)$, maximum. (c) $(0, 1)$.
**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.
(a) Chain 1: $y = 3f(x) + 2$. Chain 2: $y = 3(f(x) + 2) = 3f(x) + 6$. (b) Not the same — stretch is applied to the constant 2 as well in Chain 2. (c) Two translations (horizontal and vertical) commute, since each acts on a different coordinate.