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3.2 Algebra and Functions (A-level only)

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Question 82

The voltage VVV (in mV) in a biological sensor over time ttt (in ms) is modeled by the function V=V(t)V = V(t)V=V(t). The curve C C\,C with equation V=V(t)V = V(t)V=V(t) is described by the following features:

  • It has a single turning point, a maximum, at (4,12)(4, 12)(4,12).
  • It crosses the vertical axis at (0,7)(0, 7)(0,7) and the horizontal axis at (10,0)(10, 0)(10,0).
  • It has a horizontal asymptote with equation V=3V = 3V=3 as t→−∞t \to -\inftyt→−∞.
a.

State the equation of the horizontal asymptote to the curve with equation y=V(−t)y = V(-t)y=V(−t).

[1]
b.

State the coordinates of the maximum turning point on the curve with equation y=V(2t)y = V(2t)y=V(2t).

[2]
c.

Given that the horizontal line V=kV = kV=k, where k k\,k is a constant, intersects C C\,C at exactly one point:

Determine the possible values for kkk.

[2]
d.

The curve C C\,C is transformed to a new curve that passes through the origin.

(i) Given that the new curve has equation y=V(t)−cy = V(t) - cy=V(t)−c, state the value of the constant ccc.

(ii) Write down an equation for another single transformation of C C\,C that also results in a curve passing through the origin.

[2]

3.2 Algebra and Functions (A-level only) Questions

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