Problems — 11.5 Transformations of Functions

1

**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) $y = x^2 \to y = 4x^2$

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2

**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 $y = f(x/2)$.

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3

**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.

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4

**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.

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5

**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 $y = x^2$.

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6

**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 $g(4)$.

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7

**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 $y = -f(x)$.

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8

**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.

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9

**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 $f$.

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10

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

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11

**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 $y = g(x)$.

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12

**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.

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