L6 TEST 1
Topics: Quadratics & Functions
1.
i. The curve 𝑦 = 5√𝑥 is transformed by a stretch, scale factor
1
2
, parallel to the 𝑥-axis.
Find the equation of the curve after it has been transformed.
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ii. Described the single transformation which transform the curve 𝑦 = 5√𝑥 to the curve
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𝑦 = (5√𝑥) − 3.
2. Functions 𝑓 and 𝑔 are defined by
𝑓: 𝑥 ↦ 3𝑥 + 2, 𝑥 ∈ ℝ
𝑔: 𝑥 ↦ 4𝑥 − 12, 𝑥 ∈ ℝ
Solve the equation 𝑓 −1 (𝑥) = 𝑔𝑓(𝑥).
3.
[4]
2
).
0
Find and simplify the equation of the translated curve.
(a) The curve 𝑦 = 𝑥 2 + 3𝑥 + 4 is translated by (
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(b) The graph of 𝑦 = 𝑓(𝑥) is transformed to the graph of 𝑦 = 3𝑓(−𝑥).
Described fully the two single transformations which have been combined to give the
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resulting transformation.
4. Sketch the graph of 𝑦 = 4𝑥 2 + 8𝑥 + 3 , stating the turning point and 𝑦 -intercept and the
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points of intersection on the 𝑥-axis.
5. The function 𝑓 is defined, for 𝑥 ∈ ℝ , by 𝑓: 𝑥 ↦ 𝑥 2 + 𝑎𝑥 + 𝑏 , where 𝑎 and 𝑏 are
constants.
(a) It is given that 𝑎 = 6 and 𝑏 = −8. Find the range of 𝑓.
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(b) It is given instead that 𝑎 = 5 and that the roots of the equation 𝑓(𝑥) = 0 are 𝑘 and
−2𝑘, where 𝑘 is a constant. Find the values of 𝑏 and 𝑘.
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(c) Show that if the equation 𝑓(𝑥 + 𝑎) = 𝑎 has no real roots then 𝑎2 < 4(𝑏 − 𝑎).
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6. The function 𝑓 is defined by 𝑓: 𝑥 ↦ −𝑥 2 + 10𝑥 − 27 for 𝑥 ∈ ℝ.
(i) Express 𝑓(𝑥) in the form −(𝑥 − 𝑚)2 + 𝑛, where 𝑚 and 𝑛 are constants.
(ii) State the range of 𝑓.
(iii)Explain why 𝑓 does not have an inverse.
The function 𝑔 is defined by 𝑔: 𝑥 ↦ −𝑥 2 + 10𝑥 − 27 for 𝑥 ≤ 𝐴, where 𝐴 is a constant.
(iv) State the largest value of 𝐴 for which 𝑔 has an inverse.
(v) When 𝐴 has this value, obtain an expression, in terms of 𝑥, for 𝑔−1 (𝑥) and state the
range of 𝑔−1 .
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[ Total Marks = 38 ]