2.2.6 · D5Functions
Question bank — Graphs of functions — plotting, reading key features
Reminders of the vocabulary you'll need: a graph is the set of all points ; the Vertical line test decides "is this even a function?"; Domain and range tell you where the curve lives and how high/low it reaches; Asymptotes are lines the curve chases but never lands on; Symmetry (even and odd functions) describes mirror or rotational balance.
True or false — justify
A vertical line can cross the graph of a function at two points if the function is complicated enough.
False. The definition of a function forbids one input having two outputs, so by the Vertical line test every vertical line hits the graph at most once — complexity is irrelevant.
A horizontal line can cross the graph of a function at many points.
True. That just means several different inputs share one output (e.g. hits at and ). Nothing in the function rule forbids repeated outputs.
The graph of eventually touches the line far to the right.
False. It approaches as but is never actually (numerator is ), so it is a horizontal asymptote it hugs but never reaches.
If a curve is symmetric about the -axis, then it is the graph of an even function.
True, provided it is a function. -axis symmetry means , the definition of even; but a full circle is -axis symmetric and is not a function.
Every parabola has exactly two -intercepts.
False. It has two, one, or zero depending on the discriminant. A tangent-to-the-axis parabola touches once; one sitting entirely above the axis crosses zero times.
A function that is increasing on and increasing on must be increasing everywhere.
False. If there is a break or asymptote at (like ), the curve can jump down across the gap, so the value just after can be smaller than a value just before — increasing on each piece is not increasing overall.
The -intercept of a graph is found by setting .
False — that is the x-intercept. The -intercept is where , giving the point . Swapping these is the single most common intercept error.
Two different functions can never share the same graph.
True in the plane, but the rule can look different (e.g. and produce the identical graph). "Same graph" means "same set of points", so the functions are equal as functions even if written differently.
Spot the error
Claim: " has its minimum where because that's the smallest sample I plotted."
The vertex is at , giving the true minimum . "Smallest sample plotted" is not the minimum; you must find the turning point, not just eyeball your point list.
Claim: " is undefined at , so I'll draw a smooth curve from straight through to ."
You cannot connect through an excluded point. At there is a vertical asymptote; the curve dives to on the left and drops from on the right — two separate branches, never joined.
Claim: "Since near , I'll just stop my pencil at and start again later."
Stopping hides the behaviour. Draw a dashed vertical line at and show the curve hugging it upward and downward — that communicates "explodes here", whereas an abrupt stop looks like the domain simply ends.
Claim: "The range of is all because the -intercept is ."
A line with nonzero slope covers every real ; the intercept is just one point on it. The range is . Only bounded curves (like a parabola) have a floor or ceiling.
Claim: "The parabola opens downward because it has a minimum."
It opens upward; the leading coefficient is . A minimum (valley) is exactly what an upward parabola has — downward parabolas have a maximum.
Claim: " crosses the -axis somewhere between its two branches."
It never crosses the -axis at all — would need the numerator to be , but it is the constant . There is no -intercept anywhere.
Claim: "This curve passes through and , so it's a valid function with two roots."
Two points with the same but different means one input has two outputs — it fails the Vertical line test and is not a function at all.
Why questions
Why do we bother finding the domain before plotting rather than just picking points?
Because points you pick blindly might be forbidden inputs (e.g. in ), and knowing the domain tells you where to leave gaps and where asymptotes must go — the shape depends on it.
Why is the vertex -coordinate also the axis of symmetry of a parabola?
Completing the square gives , and is unchanged if you replace by — equal distances left and right of give equal , which is exactly mirror symmetry about .
Why does an odd function's graph look the same after a rotation about the origin?
Odd means : the point has a partner , and rotating any point about the origin sends — so every point lands on another point of the graph.
Why can reading "increasing/decreasing" straight off a graph beat doing algebra?
The eye pattern-matches "uphill vs downhill" instantly across the whole domain, whereas algebra (or the derivative) checks one region at a time; the graph shows all turning points and trends at once.
Why does a rational function have two branches rather than one connected curve?
The vertical asymptote splits the domain into two disjoint intervals ( and for ); the function is defined separately on each, and nothing connects them across the forbidden value, so the picture is genuinely two pieces.
Why do we sample points on both sides of an asymptote instead of just one?
The two sides can behave completely differently — near , goes to on the left but on the right. Sampling one side would let you falsely assume symmetry and draw the wrong branch.
Why is "degree polynomial has at most turning points" only an upper bound?
Turning points can coincide or fail to appear (e.g. is degree but has zero turning points — its slope only flattens, never reverses). The degree caps the maximum but does not guarantee that many.
Edge cases
What does the graph of a constant function look like, and is it increasing, decreasing, or neither?
A horizontal line at height . It is neither increasing nor decreasing — for any the outputs are equal, satisfying neither strict inequality.
Is increasing on all of because "squaring gets bigger"?
No. It decreases on (e.g. ) and increases on . Intuition about positive silently ignores the left half.
Can a single graph have a hole and pass the vertical line test?
Yes. A removable hole (one missing point, like at ) simply means that one has no output — every vertical line still hits at most once, so it is still a function.
What happens to the graph of for negative , and how do you show it?
Nothing — there is no curve there, since the domain is . You leave the entire left half of the plane blank and let the curve start at the origin .
For , what is the output exactly at ?
There is no output; is excluded from the domain (division by zero). The graph has an asymptote there, not a point — never write , since is not a number the function outputs.
If a horizontal line is a horizontal asymptote as , must the curve also approach it as ?
Not necessarily. A curve can flatten toward on the right yet head to on the left; each end's behaviour is independent and must be checked separately.
Is the origin always on the graph of an odd function?
Only if is in the domain. If it is, then forces , so the curve passes through the origin. But an odd function like excludes entirely, so it does not touch the origin.
Recall One-line survival kit
Domain first → mark gaps/asymptotes → sample both sides of trouble spots → connect only within continuous pieces → read features (intercepts, turning points, end behaviour) from the finished picture.