Check the graph and its symmetries

Set up the calculator, then check the mathematics

A calculator gives a view of a graph, not the whole graph. Use it to explore, then check the domain, intercepts, symmetry, turning points and asymptotes mathematically.

This lesson develops sketches of rational curves such as $(ax+b)/(cx+d)$ and $(ax^2+bx+c)/(dx+e)$. The source first introduces bar charts, pie charts, histograms and line graphs as ways to communicate numerical data before moving to Cartesian curves.

Choose the calculator mode

In MODE, use the arrow keys and ENTER to select FUNCTION. The Teacher recommends a TI-SmartView demonstration of the settings; the reconstructed menu below makes that demonstration available beside the basic graph entries. Calculator models may label operations differently. Use the dedicated negation (−) key for $e^{-x}$, not the subtraction operation.

All nine source rows are illustrated below: $x^2$, $x^3$, $\sqrt x$, $1/x$, $1/x^2$, $\sqrt[3]x$, $e^x$, $e^{-x}$ and $\ln x$. The fraction shortcut ALPHA → Y= opens FRAC: 1:n/d, 2:Un/d, 3:n/d ↔ Un/d, 4:F ↔ D. Select the numerator/denominator template for $1/x$ or $1/x^2$. For the cubic, ZOOM → 2:Zoom In → ENTER, centred at the origin, reveals the local shape.

Check what the window hides

The logarithm illustrates a window limitation: a sampled screen may give a misleading impression near $x=0$. Its real domain is $x>0$ and its vertical asymptote is $x=0$; use that mathematics when interpreting the screen.

The visible source zoom menu includes 2:Zoom In, 3:Zoom Out, 4:ZDecimal, 5:ZSquare, 6:ZStandard, 7:ZTrig, 8:ZInteger, 9:ZoomStat, 0:ZoomFit. The reconstructions use independently chosen windows, not an exact calculator emulator. The source’s “Diagrams” reminder concerns Cartesian relations and conventional drawing of asymptotic behaviour.

Check whether numerator and denominator share a factor before assigning a vertical asymptote. A removable discontinuity is a hole, and an asymptote can sometimes be crossed away from a singularity.

Find intercepts without losing domain restrictions

An axial intercept is a point where the curve meets an axis. For an x-intercept set $y=0$; for a y-intercept set $x=0$, provided that input belongs to the domain.

Example 1 shows the full six-step calculator zero procedure and five-step window/value procedure. Compare the default window with Ymin=$-20$: a hidden minimum is a display limitation. Use algebra for exact coordinates and verify numerical roots before rounding. Self-Practice 1 applies this to four rational curves with their original denominator restrictions.

Distinguish the three kinds of asymptote

An asymptote describes limiting behaviour. The arrow $\to$ means approaches. A vertical line $x=k$ is an asymptote when the curve becomes unbounded as $x$ approaches $k$ from at least one side. For these rational functions, an excluded input with nonzero numerator provides such behaviour; a cancelled factor may instead leave a hole.

A horizontal asymptote $y=k$ means $y\to k$ as $x\to+\infty$ or $-\infty$; a curve may cross it, as the source points ahead to Examples 8(c)(d).

An oblique asymptote $y=mx+c$, $m\ne0$, means $y-(mx+c)\to0$ at an infinite end.

Determine and draw dotted asymptotes before the curve. Use analytical limits to support the sketch: the Teacher warns that the calculator alone has limited capabilities for finding them.

Use division and calculus to check the screen

The source highlights long division for improper rational functions. Equal numerator/denominator degrees give a constant polynomial part, illustrated by $(3x-7)/(x+1)$ and $(x^2-1)/(5x^2+4)$; one extra numerator degree gives a sloping linear part, illustrated by $(x^2-3x)/(x+1)$ and $(x^3-3x)/(x^2-1)$. Independently their asymptotes are respectively $y=3$, $y=1/5$, $y=x-4$ and $y=x$. Self-Practice 3 retains each printed division and asymptote.

Turning points can be found using differentiation or a calculator. Example 4 now retains the complete quotient-rule calculation, second-derivative classification and six-step numerical maximum/minimum procedure. Exact calculus methods are developed further in Applications of Differentiation. Verify numerical coordinates and keep a search bracket on one continuous branch.

Decide what a missing point means

Two screens appear to show a break at $x=1$. The rules are

$a(x)=\frac{x^2-1}{x-1},\qquad b(x)=\frac{x+1}{x-1}.$

For each rule, decide whether the break is a hole or a vertical asymptote. Retain the original domain and explain what algebra establishes that changing the window alone cannot.

After your attempt: compare the reasoning

Both original domains exclude 1. For $x\ne1$, $a(x)=x+1$, so its missing point is $(1,2)$ and nearby values remain finite. It has a hole, not a vertical asymptote. For $b(x)=1+2/(x-1)$, the one-sided values become unbounded near 1, so $x=1$ is a vertical asymptote. A sampled screen can hide a single missing point or make an asymptotic branch misleading; the rule and its domain settle the distinction.

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.