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Reciprocal and exponential graphs: asymptotes

Asymptotes Extended 13 min read

On this page
  1. In this note
  2. Formula bank
  3. What an asymptote is
  4. Finding the asymptotes
  5. Which quadrants do the branches go in?
  6. Sketching any reciprocal graph
  7. Domain and range from the asymptotes
  8. Exponential graphs
  9. Ways they test it
  10. Mistakes to avoid
  11. Summary
  12. Practice
  13. Can I do this?
  14. Quiz

In this note

Syllabus 2.10/2.11 (graphs of functions): asymptotes
  • 2.10Find the vertical and horizontal asymptotes of \(y=\dfrac a{x-b}+c\) and \(y=ar^{x}+b\).
  • 2.10Tell which quadrants the branches of a reciprocal graph go in, and sketch it.
  • 2.10Read the domain and range straight from the asymptotes.
  • 2.10Find intercepts, and unknown constants from the asymptotes and a point.

Before you start: domain and range. Drawing graphs from tables: graphs of functions.

Formula bank

Reciprocal graphLearn it\(y=\dfrac a{x-b}+c\)vertical asymptote \(x=b\); horizontal asymptote \(y=c\); the asymptotes cross at \((b,\ c)\).Use it when you are given a fraction with \(x\) on the bottom.
Which quadrantsLearn it\(a>0\): top right + bottom left. \(a<0\): top left + bottom rightmeasured from the point \((b,\ c)\) where the asymptotes cross.Use it when you sketch a reciprocal graph.
Exponential graphLearn it\(y=ar^{x}+b\)horizontal asymptote \(y=b\); y-intercept \(a+b\); no vertical asymptote.Use it when the \(x\) is in the power.
Domain and range (reciprocal)Learn itdomain \(x\ne b\); range \(y\ne c\)the asymptotes are the only values never used / never reached.Use it when asked for the domain or range.
Domain and range (exponential)Learn itdomain: all \(x\); range \(y>b\) (if \(a>0\)) or \(y<b\) (if \(a<0\))the curve stays on one side of its asymptote.Use it when asked for the range of an exponential.

What an asymptote is

y = 1/x-5-4-3-2-112345-5-4-3-2-112345xyy = 1/x
y = 1/x: two separate branches.

An asymptote is a line the curve gets closer and closer to but never reaches. Every one has a reason:

Vertical, \(x=0\): you cannot divide by 0. Near 0, \(\tfrac1x\) is huge: \(\tfrac1{0.001}=1000\).

Horizontal, \(y=0\): when \(x\) is huge, \(\tfrac1x\) is tiny: \(\tfrac1{1000}=0.001\). It never quite reaches 0.

Finding the asymptotes

General methodHow to find the asymptotes of \(y=\dfrac a{x-b}+c\)

  1. Vertical: put the bottom of the fraction equal to 0 and solve. \(x-b=0\) gives \(x=b\).
  2. Horizontal: when \(x\) is huge the fraction is about 0, so \(y\) is about the number added on: \(y=c\).
  3. Write each one as the equation of a LINE: \(x=\dots\) and \(y=\dots\).

Example Write down the asymptotes of \(y=\dfrac4{x+1}-2\).

Bottom = 0 gives the vertical one; the number added on gives the horizontal one.

  1. \(x+1=0\Rightarrow x=-1\)
  2. the number added on is \(-2\)

Answer \(x=-1\) and \(y=-2\)

Try it yourself

1
Vertical asymptote of \(y=\dfrac2{x-5}+3\): \(x=\)
Show answer

Answer \(x=5\)

2
Horizontal asymptote of \(y=\dfrac2{x-5}+3\): \(y=\)
Show answer

Answer \(y=3\)

3
Vertical asymptote of \(y=\dfrac7{x+4}\): \(x=\)
Hint
\(x+4=0\).
Show answer

Answer \(x=-4\)

Which quadrants do the branches go in?

The two asymptotes cross at the point \((b,\ c)\) and cut the grid into four quarters around it. A reciprocal graph always has its two branches in OPPOSITE quarters. The sign of \(a\) tells you which pair:

y = 2/x, a positive-5-4-3-2-112345-5-4-3-2-112345xy
\(a>0\) (here \(y=\tfrac2x\)): top right and bottom left.
y = −2/x, a negative-5-4-3-2-112345-5-4-3-2-112345xy
\(a<0\) (here \(y=-\tfrac2x\)): top left and bottom right.

Why? Just to the right of \(x=b\), the bottom \(x-b\) is a tiny positive number, so \(\dfrac a{x-b}\) is huge with the SAME sign as \(a\). If \(a\) is positive, the curve shoots UP on the right of the vertical asymptote (top right), and by symmetry DOWN on the left (bottom left).

General methodHow to decide the quadrants for any reciprocal graph

  1. Write it as \(y=\dfrac a{x-b}+c\). A minus in front counts: \(y=2-\dfrac4{x+1}\) has \(a=-4\).
  2. Mark the centre \((b,\ c)\) where the asymptotes cross.
  3. \(a\) positive: top right and bottom left of the centre. \(a\) negative: top left and bottom right.
  4. Check with one point: put \(x=b+1\). You get \(y=a+c\), which is above \(c\) when \(a>0\).

Example Which quarters does \(y=2-\dfrac4{x+1}\) go in?

Rewrite to find \(a\), then use its sign.

  1. \(y=\dfrac{-4}{x+1}+2\): \(a=-4\), \(b=-1\), \(c=2\)
  2. centre \((-1,\ 2)\); \(a<0\)
  3. check \(x=0\): \(y=2-4=-2\), below 2 on the right ✓

Answer top left and bottom right of \((-1,\ 2)\)

Graphs of \(y=\dfrac a{x^{2}}\): \(x^{2}\) is never negative, so both branches are on the SAME side: both above the \(x\)-axis when \(a>0\), both below when \(a<0\). Asymptotes \(x=0\) and \(y=0\) again.

Try it yourself

1
\(y=\dfrac5{x-1}\). Is \(a\) positive or negative, and which quarters?
Show answer

Answer positive: top right and bottom left of (1, 0)

2
\(y=3-\dfrac6{x}\). Which quarters?
Hint
The minus in front belongs to a.
Show answer

Answer \(a=-6\): top left and bottom right of (0, 3)

Sketching any reciprocal graph

General methodHow to sketch \(y=\dfrac a{x-b}+c\)

  1. Draw the asymptotes \(x=b\) and \(y=c\) as DASHED lines and label them.
  2. Pick the two quarters from the sign of \(a\) (section above).
  3. y-intercept: put \(x=0\). x-intercept: put \(y=0\) and solve.
  4. Plot one more point on each side of \(x=b\) if you need a guide (e.g. \(x=b+1\) and \(x=b-1\)).
  5. Draw each branch as a smooth curve through its points, bending towards both asymptotes and never touching them.
y = 3/(x − 2) + 1-4-3-2-11234567-5-4-3-2-11234567xy(0, −0.5)(−1, 0)(3, 4)
\(y=\tfrac3{x-2}+1\)

Example Sketch \(y=\dfrac3{x-2}+1\).

Asymptotes, quarters from \(a=3\), intercepts, then draw.

  1. asymptotes \(x=2\), \(y=1\); centre \((2,\ 1)\)
  2. \(a=3>0\): top right and bottom left
  3. \(x=0\): \(y=\tfrac3{-2}+1=-0.5\)
  4. \(y=0\): \(\tfrac3{x-2}=-1\Rightarrow x=-1\)
  5. check \(x=3\): \(y=4\) (top right ✓)

Answer the curve on the left

Try it yourself

1
For \(y=\dfrac6{x-3}+1\), find the y-intercept: \(y=\)
Show answer

Answer \(y=-1\)

2
For \(y=\dfrac6{x-3}+1\), find the x-intercept: \(x=\)
Hint
\(\tfrac6{x-3}=-1\), so \(x-3=-6\).
Show answer

Answer \(x=-3\)

Domain and range from the asymptotes

The domain is every \(x\) you may put in; the range is every \(y\) that comes out. A sketch shows both at a glance: the curve covers every \(x\) except the vertical asymptote, and every \(y\) except the horizontal one.

General methodHow to find the domain and range of any reciprocal or exponential graph

  1. Sketch it (even roughly): asymptotes first.
  2. Domain: all \(x\) except where it is undefined. Reciprocal: \(x\ne b\). Exponential: all real \(x\).
  3. Range: look up and down the graph. Reciprocal: every \(y\) except \(y=c\), so \(y\ne c\). Exponential: only one side of \(y=b\), so \(y>b\) (or \(y<b\) if \(a\) is negative).
  4. If the question gives a restricted domain (e.g. \(x>2\)), use only that part of the sketch and read the \(y\) values it reaches.

Example State the domain and range of \(y=\dfrac4{x+1}-2\).

Read them off the asymptotes.

  1. asymptotes \(x=-1\), \(y=-2\)
  2. domain: \(x\ne-1\)
  3. range: \(y\ne-2\)

Answer domain \(x\ne-1\), range \(y\ne-2\)

Going on to A Level. In 9709 Paper 1 you find the range of a function on a restricted domain all the time. The method is exactly step 4: sketch, take the part of the curve you are allowed, read off the lowest and highest \(y\). See also domain and range.

Try it yourself

1
Range of \(y=\dfrac3{x-2}+1\): \(y\ne\)
Show answer

Answer \(y\ne1\)

2
Range of \(y=2^{x}-3\): \(y>\)
Hint
The curve stays above its asymptote.
Show answer

Answer \(y>-3\)

Exponential graphs

y = 2^x − 3-4-3-2-11234-4-3-2-1123456xy(0, −2)
y = 2ˣ − 3: horizontal asymptote y = −3.

\(2^{x}\) is always positive and tends to 0 as \(x\) gets very negative (\(2^{-10}\approx0.001\)). So \(y=2^{x}\) has asymptote \(y=0\), and \(y=2^{x}-3\) is moved down 3: asymptote \(y=-3\).

General methodHow to sketch \(y=ar^{x}+b\)

  1. Draw the asymptote \(y=b\) (dashed). There is no vertical asymptote.
  2. y-intercept: \(x=0\) gives \(y=a+b\).
  3. Growth (\(r>1\)): the curve rises to the right and flattens onto \(y=b\) on the left. Decay (\(0<r<1\)): the other way round.
  4. x-intercept if there is one: put \(y=0\) and solve \(ar^{x}=-b\).

Example For \(y=3\times2^{x}-6\), find the asymptote and where it crosses both axes.

Asymptote = the number added on; then x = 0 and y = 0.

  1. asymptote \(y=-6\)
  2. \(x=0\): \(y=3-6=-3\)
  3. \(y=0\): \(3\times2^{x}=6\Rightarrow2^{x}=2\Rightarrow x=1\)

Answer \(y=-6\); \((0,\ -3)\) and \((1,\ 0)\)

Ways they test it

Type 1 Write down the asymptotes

"Write down the equations of the asymptotes of \(y=\dfrac4{x+1}-2\)." (2 marks)

Spot it the words "equations of the asymptotes".

  1. Bottom = 0 for the vertical one.
  2. The number added on for the horizontal one.
  3. Answer as \(x=\dots\), \(y=\dots\).

Type 2 Sketch the graph

"Sketch \(y=\dfrac3{x-2}+1\), showing the asymptotes and where it crosses the axes." (3 marks)

Spot it "sketch" with a reciprocal or exponential equation.

  1. Asymptotes dashed and labelled.
  2. Branches in the right quarters (sign of a).
  3. Intercepts marked with their values.

Type 3 Find the constants

"\(y=\dfrac a{x-b}+c\) has asymptotes \(x=1\), \(y=2\) and passes through (3, 5). Find \(a\), \(b\), \(c\)."

Spot it unknown letters in the equation + asymptotes or a point.

  1. \(b\) and \(c\) come straight from the asymptotes.
  2. Put the point in to find \(a\).

For example asymptotes \(x=1\), \(y=2\); through (3, 5)

  1. \(b=1\), \(c=2\)
  2. \(5=\tfrac a{3-1}+2\Rightarrow a=6\)

Answer \(a=6\), \(b=1\), \(c=2\)

Type 4 Domain and range

"State the range of \(y=2^{x}-3\)."

Spot it the words domain, range, or "values that must be excluded".

  1. Find the asymptotes.
  2. Domain: leave out the vertical one. Range: leave out the horizontal one (exponential: one side only).

Type 5 Match the graph

"Which of these sketches shows y = −2/x?" (or match equations to graphs)

Spot it several graphs and equations.

  1. Find each asymptote.
  2. Use the sign of a for the quarters; for an exponential, the intercept a + b.

Mistakes to avoid

  • ✗ \(y=\dfrac4{x+1}\): asymptote \(x=1\)✓ \(x=-1\)Solve \(x+1=0\).
  • ✗ \(y=2-\dfrac4{x+1}\): \(a=4\), so top right✓ \(a=-4\): top left and bottom rightThe minus in front belongs to a.
  • ✗ Asymptotes given as "−1 and −2"✓ \(x=-1\) and \(y=-2\)An asymptote is a line: give its equation.
  • ✗ Branches touching or crossing the dashed lines✓ branches bend towards them and never meet themThat is what an asymptote means.
  • ✗ Range of \(y=\dfrac3{x-2}+1\): \(y\ne2\)✓ \(y\ne1\)Range uses the HORIZONTAL asymptote (\(y=c\)); domain uses the vertical one.

Summary

  • \(y=\dfrac a{x-b}+c\): asymptotes \(x=b\) and \(y=c\), crossing at \((b,\ c)\).
  • \(a>0\): branches top right and bottom left; \(a<0\): top left and bottom right.
  • \(y=ar^{x}+b\): one asymptote \(y=b\); y-intercept \(a+b\).
  • Domain leaves out the vertical asymptote; range leaves out the horizontal one.
  • Sketch: dashed asymptotes, the right quarters, intercepts with values, curves that never touch the dashes.

If you see …

  • a fraction with x on the bottombottom = 0 gives the vertical asymptote.
  • a minus in front of the fractiona is negative: top left and bottom right.
  • "range"sketch first, then read the y values the curve reaches.
  • unknown a, b, c and a pointb, c from the asymptotes, then substitute the point.

Practice

Practice: easy, medium, hard

Easy

1
Write down the vertical asymptote of \(y=\dfrac5{x-1}\): \(x=\)
Hint
Bottom = 0.
Show working
  1. \(x-1=0\)

Answer \(x=1\)

2
Write down the horizontal asymptote of \(y=\dfrac3{x}+2\): \(y=\)
Hint
The number added on.
Show working
  1. the fraction tends to 0, so \(y\to2\)

Answer \(y=2\)

3
Find the y-intercept of \(y=5\times2^{x}+1\).
Hint
Put \(x=0\): \(2^{0}=1\).
Show working
  1. \(5\times1+1=6\)

Answer \(y=6\)

Medium

1
For \(y=\dfrac2{x+3}+1\), find where the curve crosses (a) the y-axis (b) the x-axis.
Hint
(b) \(\tfrac2{x+3}=-1\).
Guide me

What is it asking? two intercepts

What do we know? \(a=2\), \(b=-3\), \(c=1\)

Plan \(x=0\) for the y-axis; \(y=0\) for the x-axis

Show working
  1. (a) \(\tfrac23+1=\tfrac53\)
  2. (b) \(\tfrac2{x+3}=-1\Rightarrow x+3=-2\Rightarrow x=-5\)

Answer (a) \(y=\tfrac53\) (b) \(x=-5\)

2
\(y=3-\dfrac6{x-2}\). Write down the asymptotes and say which quarters the branches are in.
Hint
Rewrite as −6/(x − 2) + 3.
Guide me

What is it asking? asymptotes and quarters

What do we know? a minus in front of the fraction

Plan read b and c, then use the sign of a

Show working
  1. \(y=\tfrac{-6}{x-2}+3\)
  2. asymptotes \(x=2\), \(y=3\)
  3. \(a<0\): top left and bottom right of \((2,\ 3)\)

Answer \(x=2\), \(y=3\); \(a=-6\): top left and bottom right

3
State the range of \(y=\dfrac1{x+4}-5\): \(y\ne\)
Hint
The horizontal asymptote.
Guide me

What is it asking? the range

What do we know? \(c=-5\)

Plan range leaves out the horizontal asymptote

Show working
  1. horizontal asymptote \(y=-5\), so \(y\ne-5\)

Answer \(y\ne-5\)

Hard

1
\(y=\dfrac a{x-2}-3\) passes through (4, 2). Find \(a\).
Hint
Substitute \(x=4\), \(y=2\).
Guide me

What is it asking? the value of \(a\)

What do we know? the point (4, 2) is on the curve

Plan substitute and solve

Show working
  1. \(2=\tfrac a{4-2}-3\)
  2. \(5=\tfrac a2\Rightarrow a=10\)

Answer \(a=10\)

PartAnswer / workingMarksGuidance
\(2=\tfrac a{2}-3\)M1
\(a=10\)A1
Total: 2 marks
2
Sketch \(y=\dfrac{-6}{x-2}+3\), showing the asymptotes and both intercepts.
Hint
Intercepts: x = 0, then y = 0.
Guide me

What is it asking? a sketch with asymptotes and intercepts

What do we know? \(a=-6\), \(b=2\), \(c=3\)

Plan dashes, quarters from the sign of a, intercepts, draw

Show working
  1. asymptotes \(x=2\), \(y=3\)
  2. \(a<0\): top left and bottom right
  3. \(x=0\): \(y=3+3=6\)
  4. \(y=0\): \(\tfrac{-6}{x-2}=-3\Rightarrow x-2=2\Rightarrow x=4\)

Answer asymptotes \(x=2\), \(y=3\); \((0,\ 6)\) and \((4,\ 0)\); branches top left and bottom right

PartAnswer / workingMarksGuidance
\(x=2\) and \(y=3\) shownB1
two branches in the correct quartersB1
\((0,\ 6)\) and \((4,\ 0)\)B1
Total: 3 marks
3
\(f(x)=\dfrac2{x-1}+4\) for \(x>1\). Find the range of \(f\).
Hint
Only the branch to the right of x = 1. Is it above or below y = 4?
Guide me

What is it asking? the range on a restricted domain

What do we know? \(x>1\) only; \(a=2>0\)

Plan sketch, keep the right-hand branch, read the y values

Show working
  1. asymptotes \(x=1\), \(y=4\)
  2. \(a>0\), so the right-hand branch is top right: above \(y=4\)
  3. it comes down towards 4 but never reaches it

Answer \(f(x)>4\)

PartAnswer / workingMarksGuidance
sketch or reasoning with the right-hand branchM1
\(f(x)>4\)A1
Total: 2 marks

Can I do this?

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Quiz

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