Transformations & asymptotes

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Consider the rational function g(x)=x−cax+b where a,b,c∈Z and x=c. The graph of y=g(x) is shown in the diagram below.

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Write down
1. the value of a,
  [1]
2. The value of c.
  [1]
Find the value of b.
[2]

2

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Consider the rational function g(x)=3+2x7−4x, for x=−23.

For the graph of y=g(x) find
1. the coordinates of the axes intercepts,
  [2]
2. the equations of the asymptotes.
  [2]
Hence sketch the graph of y=g(x) on the axes below.
[2]

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The following diagram shows the graph of a function f.

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On the axes above, sketch the graph of y=2f(x+3)−1, indicating clearly the point A′ corresponding to point A.
[4]

Consider a function g defined by g(x)=af(x)+b. The graph of g can be obtained from the graph of f by

A vertical translation of 2 units followed by
A vertical stretch with scale factor 3.

Find the value of a and the value of b.
[3]

4

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Consider the rational function f(x)=kx+5ax+b, for x=−k5, where a,b,k∈Z. The graph of y=f(x) has asymptotes at y=5 and x=35, and passes through the point Q(2,25).

Find the values of a,b and k.

[6]

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IB: 5

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The graph of y=g(x) is shown in the diagram below.

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On the set of axes below, sketch the graphs of

y=[g(x)]2,
[3]
y=g(∣x∣)1.
[4]

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The function g is defined by g(x)=5x+86−5x, for x=−58.

Sketch the graph of y=g(x), labeling any asymptotes or axes intercepts.
[5]
Find an expression for g−1(x).
[3]

The graphs y=g(x) and y=g−1(x) intersect at two points.

Find the coordinates of the points of intersection.
[5]

9

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Consider the quadratic function f(x)=−3x2+18x−15.

The function f can be expressed in the form f(x)=a(x−h)2+k.

Find the values of a, h and k.
[4]
Hence find, on the graph of y=f(x),
1. the coordinates of the vertex,
  [1]
2. the axes intercepts.
  [4]

The graph of g can be obtained from the graph of f by

A vertical stretch by scale factor −32 followed by
A translation of (−47)

Find g(x).
[4]

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