Check combined transformations

Track retained, duplicated and excluded labelled points

First decide whether a transformation moves every point, retains only half of a graph, or excludes an input. Then apply the rule to each labelled point separately. A correct outline is not enough if a requested point has been lost.

Self-Practice 7 applies the reciprocal rule to $x+4$: the independent sketch has pole $x=-4$, horizontal asymptote 0 and y-intercept $1/4$. Example 19 combines the earlier methods in eight separate sketches, preserving all intermediate stages. Use a coordinate map for the current equation, as the Teacher recommends:

Apply each target to an original point (p, q)

$f(x-2)$

$(p,q)\mapsto(p+2,q)$.

$f(-x)+1$

$(p,q)\mapsto(-p,q+1)$.

$f(2x+1)$

$(p,q)\mapsto((p-1)/2,q)$.

$2f(\lvert x\rvert)$

Retain $p\ge0$, duplicate at $x=\pm p$ and double the output.

$f(-\lvert x\rvert)$

Retain $p\le0$ and duplicate at $x=\pm\lvert p\rvert$.

$\lvert f(x-2)\rvert$

$(p,q)\mapsto(p+2,\lvert q\rvert)$.

$f(\lvert x-2\rvert)$

Retain $p\ge0$, duplicate at $x=2\pm p$; symmetry axis $x=2$.

$1/f(x)$

Retain only non-zero outputs, using $(p,1/q)$.

Count images and check exclusions

A point on the retained fold line has coincident images and is counted once.

Discarded original inputs have no corresponding point in an input-modulus construction; other points may have two images.

Keep every requested A–E correspondence rather than listing only surviving turning points.

For part (c), the Teacher notes that the method is the same as Example 8 and need not be retaught in detail; reuse its replacements while retaining all coordinates here.

Keep the precision of the given coordinates

The given B and D coordinates are printed approximations. Transform these consistently without inventing extra precision. In part (b), reflection keeps B’s x-magnitude $0.549$; the source’s final label 0.5 is a typo, retained in native/review evidence. In part (d), the final heights 4.226 confirm the factor 2 omitted from the source final curve label.

Parts (f) and (g) distinguish taking output modulus after a shift from shifting an input-modulus graph. In part (h), original roots A/C/E become poles, while B/D acquire reciprocal heights about $-1.585$ and $0.473$. All source stages, including the original graph repeated in part (h), remain available.

Assess the source mastery objectives twice

The source provides separate self-assessment columns at the end of the lecture and at the end of the tutorial. Check each objective at both stages; reading a solution does not establish mastery. From $y=f(x)$:

• Obtain/describe $y=f(x-a)$.
• Obtain/describe $y=f(x+a)$.
• Obtain/describe $y-a=f(x)$.
• Obtain/describe $y+a=f(x)$.
• Obtain/describe $y=f(ax)$.
• Obtain/describe $ay=f(x)$.
• Obtain/describe $-y=f(x)$.
• Obtain/describe $y=f(-x)$.
• State the variable replacement required for translation, scaling and reflection.
• Describe a suitable sequence of transformations.
• Obtain $y=\lvert f(x)\rvert$.
• Obtain $y=f(\lvert x\rvert)$.
• Obtain $y=1/f(x)$, retaining its domain exclusions.

Use the complete source study map

The printed references below identify the source’s intended practice sequence. The closing Discussion 1–9 and Review 1–9 have now been compared with the existing canonical records; their two lecture-copy appearances retain independent pagination through source aliases, without creating duplicate exercises. The lesson’s Practice, Discussion and Review navigation opens those canonical exercises.

The prior-topic rows retain Graphing Techniques → Discussion 5(i)(ii)(iii),7(i)(ii)(iii)(a), and Functions (Piece-wise) → Discussion 9(ii)(iii). The Graphing Techniques reference is reproduced as printed rather than silently resolving its compressed subpart notation. The source row says “Identity error”; the teaching label uses “Identify and correct” while retaining the original wording in the native transcription.

Closing source appearances and printed qualifications

The closing question asks, graph features, attributions and Review mark allocations match the retained canonical Discussion/Review exercises. Teacher PDF 34 / printed 106 is a blank body page, so later question pages differ between copies.

The short keys answer only the source-listed items; the canonical worked solutions retain their fuller standalone-solution provenance.

Discussion 8 uses $f(3-x/2)$, not the expression suggested by flattened fraction extraction; Discussion 7 asks for an absolute reciprocal.

Review 1’s given B diagram label(2,0) conflicts with its prose(0,2); the canonical coordinate follows the prose and y-axis position.

Review 5’s original graph, absent from the standalone solution PDF, is present in both lecture copies and now directly confirms the reconstruction.

Original typos and reciprocal-domain qualifications remain recorded rather than silently erased.