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Further Graphs and Tangents

Coordinate Geometry and Graphs

Lines the curve approaches but never reaches

  • An asymptote is a straight line that a curve gets closer and closer to without ever touching
  • Two of them belong to y = a ÷ x: the y-axis, x = 0, and the x-axis, y = 0
    • The vertical one comes from the value that would make the bottom zero
    • The horizontal one is where the curve settles as x grows very large in either direction
  • Adding a constant lifts the whole curve, so y = a ÷ x + b has the horizontal asymptote y = b while the vertical one stays at x = 0
The curve y = 1/x with both of its asymptotes drawn as dashed lines and labelled: the horizontal asymptote y = 0 along the x-axis and the vertical asymptote x = 0 along the y-axis, with the two branches approaching each without touching
Source: Types of Graphs by Save My Exams
  • The curve y = a ÷ x² is always positive and both branches sit above the x-axis, unlike y = a ÷ x whose branches sit in opposite quadrants
Two sketches of y = a/x showing what the value of a does: with a far from zero the branches sit well away from the axes, and with a close to zero they are pulled in tight against both axes
Source: Types of Graphs by Save My Exams
Two sketches on separate axes: y = 1 over x with branches in opposite quadrants, and y = 1 over x squared with both branches above the x-axis, each approaching the axes without touching them.
Source: Types of Graphs by Save My Exams

Exponential curves

  • The curve y = kˣ passes through (0, 1) whatever k is, and has the single asymptote y = 0
  • It grows when k is greater than 1 and decays towards zero when k lies between 0 and 1
  • Stretching and shifting gives y = akˣ + b, which meets the y-axis at (0, a + b) and has asymptote y = b
  • In a growth or decay context, b is the value the quantity settles at in the long run and a + b is its starting value
  • Asymptotes are written as equations of lines, so the answer is x = 0 or y = 3, never a bare number
Worked example

Writing down the asymptotes

Write down the equations of the asymptotes of y = 6/x, and of y = 2ˣ + 3.

Solution:

  • An asymptote is a line the curve approaches but never meets, and it is written as an equation, not a number
  • For y = 6/x, no value of y comes from x = 0, because dividing by zero is undefined
  • So the curve never crosses the vertical line x = 0, which is the y-axis
  • However large x becomes, 6/x shrinks towards zero without reaching it
  • So the curve never meets y = 0, which is the x-axis
  • For y = 2ˣ + 3, the 2ˣ part shrinks towards zero as x becomes large and negative, but never reaches it
  • The whole curve is therefore always just above 3, giving the asymptote y = 3
  • Answering "0" or "3" alone scores nothing: an asymptote is a line and needs x = or y = in front of it