Transform and verify the whole curve

Reflect negative outputs for a modulus

Absolute value makes an output nonnegative without changing its magnitude. The source compares $y=x$ with $y=|x|$ at these seven inputs:

• Inputs $-3$ and $3$ both give output $3$.
• Inputs $-2$ and $2$ both give output $2$.
• Inputs $-1$ and $1$ both give output $1$.
• Input $0$ gives output $0$.

Reflect only the part below the x-axis

To obtain $y=|f(x)|$ from $y=f(x)$, reflect just the points where $f(x)<0$ in the x-axis; retain every other section.

The reflected section has equation $y=-f(x)$ because each output changes sign. Thus $|x|=x$ for $x\ge0$ and $|x|=-x$ for $x<0$. The source’s line/V diagrams and labelled piecewise V are reconstructed below.

Its second illustration, $|x^2-3x+2|$, reflects only the negative section between the roots 1 and 2, preserving all points outside that interval and the zeros themselves.

Solve each modulus section on its own domain

Example 10 links the source’s “Equivalence” idea—algebraic and graphical representations support one another—to solving a modulus intersection.

Use the Cartesian equation valid on each section and reject candidates outside that section. Calculator intersect commands provide a numerical check; exact questions still require algebraic/analytical values.

The complete six-step calculator sequence and all source screen readings appear beside the example’s working.

Self-Practice 4 completes the preceding rational-curve practice; only its part (iii) turning points are printed answers, and independent solutions of the other asks are identified explicitly.

Further examples: verify the whole curve and solution domain

Use this final check before accepting a sketch or a root count:

1. Check the original domain, even after cancellation or a discriminant calculation.
2. Inspect both infinite ends and every continuous branch; a distant turning point may lie outside the displayed window.
3. Distinguish an intersection’s x-coordinate from its y-coordinate.
4. Count distinct admissible solutions, not just visible crossings.

What to check in Examples 11–13

Examples 11–13 combine parameter recovery, complete rational sketches, algebraic output ranges and graph intersections. An asymptote can be crossed, and a finite sketch may omit a distant stationary point: Example 11 independently recovers a far-left minimum absent from the source drawing, with its original maximum label and source excerpt retained.

Example 12 uses intersection x-coordinates to count quartic roots; the Teacher specifically distinguishes these inputs from the intersection ordinates.

Example 13 checks both the original excluded input after a discriminant argument and the distinct real solutions of a line through the asymptote intersection.

Mastery: revisit after the lecture and after the tutorial

The source leaves separate blank columns for self-assessment at these two checkpoints. Use the following full checklist at each stage; it records goals, not a claim that they are mastered:

• Features of graphs: find axial intercepts, asymptotes, turning points, lines of symmetry and possible ranges of input $x$ and output $y$.
• Using a graphing calculator: obtain a function’s graph with Zoom In, ZBox and appropriate window settings; obtain axial intercepts and stationary-point coordinates where they exist.
• Using an analytical method: determine asymptotes, axial intercepts and stationary-point coordinates where they exist.

Use the worked source examples to compare numerical readings with algebraic evidence. The common-question map below connects these objectives to the lecture examples and the repeated Discussion and Review questions.