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TMUA 2020 · Paper 2 · Question 5 of 20

TMUA 2020 Paper 2 Question 5

Graphs and transformations — Recognising a graph from its asymptotes. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Graphs and transformationsRecognising a graph from its asymptotes6 options

The original question includes a diagram: Six sketched graphs labelled A to F, each on unlabelled x and y axes, with dotted lines marking horizontal asymptotes.

Which one of the following shows the graph of y = 2x1 + 2x

(Dotted lines indicate asymptotes.)

The six sketches are:

A   an increasing curve that runs along the x-axis on the far left and flattens out onto a dotted horizontal asymptote at a positive height on the right
B   an increasing curve rising from near the x-axis with no upper asymptote drawn
C   an increasing curve rising without limit, sitting above a dotted horizontal asymptote at a positive height that it approaches on the far left
D   a decreasing curve that starts on a dotted horizontal asymptote at a positive height on the left and flattens onto the x-axis on the right
E   a decreasing curve falling towards the x-axis, with no asymptote drawn other than the axis
F   a decreasing curve falling from the top left onto a dotted horizontal asymptote at a positive height

  1. AA
  2. BB
  3. CC
  4. DD
  5. EE
  6. FF
Show the answer and worked solution
answer · A
  1. AA
  2. BB
  3. CC
  4. DD
  5. EE
  6. FF
Read off the two ends before anything else. As x → −∞, 2x → 0, so y → 0: the curve hugs the x-axis on the left. As x → +∞, divide top and bottom by 2x to get y = 12−x + 1 → 1, so there is a horizontal asymptote at y = 1 on the right. The numerator and denominator are both positive and the denominator is the larger, so 0 < y < 1 throughout and the curve increases (at x = 0 it passes through 12). Only sketch A rises from the x-axis to a horizontal asymptote above it.