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Transformations of Functions · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQTranslation, reflection, stretching and compression7 marksPaper 3~11 min

The quadratic function

f

is defined by

f(x) = (x-2)^2 - 1

, for

x \in \mathbb{R}

. Its graph is a parabola with vertex

V(2, -1)

(a)

The function

g

is obtained by reflecting

f

in the

y

-axis and then translating the result by

\begin{pmatrix} 2 \\ -1 \end{pmatrix}

. Show that

g(x) = x^2 - 2

. [3 marks]

(b)

The function

h

is defined by

h(x) = f(2x) + 1

. (i) Show that

h(x) = 4(x-1)^2

. [2]
(ii) Determine a sequence of two transformations applied to

f

that produces

h

. State each transformation precisely. [2 marks]

(c)

The claim is made: "For any quadratic function $f(x) = a(x-h)^2 + k$ , reflecting the graph in the $x$ -axis maps the vertex $V(h,\, k)$ to the point $(h,\, -k)$ ." (i) Verify this claim for

f(x) = (x-2)^2 - 1

. [1]
(ii) Prove the claim for the general case

f(x) = a(x-h)^2 + k

, where

a \neq 0

. [2 marks]

Solutions

Questions 2-8

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