IB Mathematics: Analysis & Approaches
Rational functions 1
Questions 1-6 (questions 4, 5 & 6 are HL only)
[answers included]
(a) Sketch the graph of the function without using a GDC. On your sketch, clearly indicate all x- and
y-intercepts and all horizontal and vertical asymptotes (and oblique asymptotes for HL questions).
(b) State the domain and range of the function.
(c) Use a GDC to check your answers for parts (a) and (b).
1.
y=
2
x −3
2.
y=
2x + 8
x−4
3.
y=
12 − 4x
x
4.
y=
x−2
HL only
x − x − 12
5.
y=
4 x2
HL only
x2 + 1
6.
y=
x2 + x − 2
HL only
x +1
7.
Consider the function g ( x ) =
2
ax + 6
where a is a real number constant.
x−a
(a) State the equation of the vertical asymptote for the graph of g.
(b) State the equation of the horizontal asymptote for the graph of g.
(c) The graph of g intersects the x-axis at point P, and intersects the y-axis at point Q. State the
coordinates of P and the coordinates of Q.
In 8-13, match each function with one of the graphs labelled (a), (b), (c), (d) and (e) shown below.
Do not use a GDC.
8.
y=
2
x +1
9.
y=
1
x−4
10.
y=
x +1
x
11.
y=
1 − 2x
x
12.
y=
x−2
x −1
13.
y=−
x+2
x +1
(a)
(b)
(c)
(d)
(e)
(f)
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IB Mathematics: Analysis & Approaches
Rational functions 1
ANSWERS
For questions 1-6, asymptotes and their equations are shown in red
1. (a)
(b) domain: x  , x  3
range: y  , y  0
2. (a)
(b) domain: x  , x  4
range: y  , y  2
3. (a)
(b) domain: x  , x  0
range: y  , y  − 4
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IB Mathematics: Analysis & Approaches
Rational functions 1
4. (a)
(b) domain: x  , x  −3, 4
range: y 
5. (a)
(b) domain: x  ; range: 0  y  4
6. (a)
(b) domain: x  , x  −1 ; range: y 
7. (a) x = a
8.
(f)
9.
(e)
(b) y = a
6
 6 

(c) P  − ,0  , Q  0, − 
a
 a 

10. (a)
11. (b)
12. (c)
13. (d)
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