Recognise curves and asymptotes

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.