Graphing techniques

Reflection and lines of symmetry

A line of symmetry must reflect the whole graph onto itself, including its domain. A shape having two similar-looking halves is not enough: check what happens to every point under the proposed reflection.

The seven source examples below use the Teacher’s completed answers; the Student answer column is blank.

Reflection axes for the seven curves

$y=x^2$

$x=0$.

$x=y^2$

$y=0$.

$y=\lvert x\rvert$

$x=0$.

$y=1/x$

$y=x$ and $y=-x$.

$y=\cos x$

$x=0$.

$x^2+y^2=25$

Any straight line through the origin, for example $x=0$ and $y=0$.

$y=x^3+x^2$

None.

The reciprocal’s two diagonal dashed lines are completed in the Teacher sketch. The cosine sketch shows heights $1$ and $-1$; $x=0$ is a valid source answer, but is not exhaustive: independently $x=k\pi$, $k\in\mathbb Z$, are all reflection axes of the full cosine graph. “None” for the cubic refers to reflection in a line; it is not a claim about every kind of symmetry.

Check the printed cubic coordinates

Both original cubic sketches label a maximum $(-0.677,0.148)$ and the intercept $(-1,0)$, as well as the origin. Independently $y'=3x^2+2x=x(3x+2)$ gives the exact maximum $(-2/3,4/27)$, approximately $(-0.667,0.148)$. The printed first coordinate is a source slip. The reconstruction uses the exact point and records the original printed label explicitly.

Test symmetry against the domain too

The source’s even-power test follows from $x^2=(-x)^2$: replacing $x$ by $-x$ leaves an equation containing only even powers of $x$ unchanged, so $x=0$ is a reflection axis. Similarly, only even powers of $y$ give the axis $y=0$. Independently, this conclusion assumes any stated domain restriction is also preserved by the reflection; the equation alone does not restore excluded points.

The following source section introduces possible ranges of $x$ and $y$: restrictions may be implicit in the equation or visible in the graph. Check both rather than assuming every real input or output is possible. The worked range examples follow next.

Find possible outputs algebraically

Example 5 compares three input/output restrictions. Logarithms require a positive argument; real square roots require a nonnegative radicand, and the explicit negative branch fixes the output sign.

For a rational curve, retain its denominator restriction and rearrange into a quadratic in the input. Requiring a nonnegative discriminant finds possible outputs; also check that the recovered inputs belong to the original domain.

The Teacher’s exact algebra and parabola argument are integrated beside that example’s working. Rounded turning points from a graph support the result but do not replace exact range endpoints.

The source next lists polynomial, exponential, logarithmic, trigonometric, inverse trigonometric, rational and modulus function families. It points to Revision (From O Levels) Section 5 for basic exponentials/logarithms and Section 7 for basic trigonometric/inverse trigonometric graphs, then focuses here on polynomial, exponential, rational and modulus functions. These are retained source cross-references, not a claim that those remaining Revision pages have been incorporated.

Polynomial and exponential shapes

A polynomial has the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$, with nonnegative integer $n$ and constant coefficients. With $a_n\ne0$ and $n>0$, $n$ is its degree. The source illustrates degree 1 with $2x+1$ and degree 2 with $x^2+x+1$. Independently, a nonzero constant has degree 0; the zero polynomial has no degree under the usual convention.

Build a polynomial sketch from its factors

1. Find the roots and their multiplicities. Odd multiplicity gives a crossing; even multiplicity gives contact.
2. Set x to zero to find the y-intercept.
3. Use the leading term to determine the behavior at both infinite ends.
4. Check any stationary points before joining the branches.

Example 6 shows why a horizontal tangent can be a turning point or a stationary inflection. Example 7 recovers a cubic: the three crossings determine three linear factors, then a fourth known point fixes the multiplicative constant.

Compare both ends of an exponential graph

For combinations of exponentials, the source builds on secondary-school graph shapes. Check the whole expression at each end, then locate intercepts and stationary points. Example 8 integrates the Teacher’s calculator-first demonstration (TI-SmartView or a graphing calculator minimum/maximum search), with differentiation available to verify the coordinates. For the quotient, divide numerator and denominator by $e^{2x}$ before taking the right-hand limit; never use an indeterminate $\infty/\infty$ as an answer.

Rational curves and calculator tables

A rational function is $P(x)/Q(x)$ for polynomials $P,Q$, with $Q(x)\ne0$ on its domain.

The source calls it proper when $\deg P<\deg Q$, and improper otherwise: $3/(x+1)$ is proper; $3x/(x+1)$ and $3x^2/(x-1)$ are improper.

Divide an improper expression into its polynomial quotient plus a proper remainder before sketching. This exposes constant or slant asymptotes in these examples; independently, higher degree quotients may give polynomial asymptotes rather than horizontal/slant lines.

Check cancellation and retain the original excluded inputs. A cancelled denominator factor does not make the original input admissible.

Use a table to make a conjecture

For the preceding exponential quotient, the source’s five-step calculator-table procedure uses Y₁, TBLSET via 2nd→WINDOW, TblStart 100 and −100 with ΔTbl 1, then TABLE via 2nd→GRAPH to examine both ends. The Teacher calls the outputs a way to guess horizontal asymptotes. The reconstructed table and transcribed procedure are in Example 8; the algebraic limits justify the conjecture, since rounding can display an asymptote value that is never attained.

Change the window, then verify the result

Example 9 puts algebra before screen appearance. ZStandard can miss a branch, make a maximum seem to lie on an axis, or make the left tail of a curve with asymptote $y=1$ seem to approach 0.

Adjust the window or use ZBox to inspect a region, then compare the coordinates and asymptotes with exact working.

Calculator illustrations are reconstructed beside the relevant parts, with original excerpts in review evidence, with accessible text descriptions.

Recover a rule from graph features

A curve is known to have the form $y=(ax+b)/(x+2)$. Its vertical asymptote is $x=-2$, its horizontal asymptote is $y=3$, and it passes through $(0,1)$. Find a possible rule and its x-intercept. Explain how you can check the side from which each branch approaches the horizontal asymptote.

After your attempt: compare the reasoning

The leading-coefficient ratio gives $a=3$, and the y-intercept gives $b/2=1$, so $b=2$. Thus

$y=\frac{3x+2}{x+2}=3-\frac4{x+2},\qquad x\ne-2.$

The x-intercept is $(-2/3,0)$. For $x>-2$, the remainder is negative, so the right branch lies below 3; for $x<-2$, it is positive, so the left branch lies above 3. The numerator is nonzero at $-2$, confirming a vertical asymptote rather than a hole.

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.