IB DP Maths Topic 2.3 Reflections (in both axes): y=−f(x) ; y=f(−x) SL Paper 1

Question

Let \(f(x) = 3{x^2} – 6x + p\). The equation \(f(x) = 0\) has two equal roots.

Write down the value of the discriminant.

[2]
a(i).

Hence, show that \(p = 3\).

[1]
a(ii).

The graph of \(f\)has its vertex on the \(x\)-axis.

Find the coordinates of the vertex of the graph of \(f\).

[4]
b.

The graph of \(f\) has its vertex on the \(x\)-axis.

Write down the solution of \(f(x) = 0\).

[1]
c.

The graph of \(f\) has its vertex on the \(x\)-axis.

The function can be written in the form \(f(x) = a{(x – h)^2} + k\). Write down the value of \(a\).

[1]
d(i).

The graph of \(f\) has its vertex on the \(x\)-axis.

The function can be written in the form \(f(x) = a{(x – h)^2} + k\). Write down the value of \(h\).

[1]
d(ii).

The graph of \(f\) has its vertex on the \(x\)-axis.

The function can be written in the form \(f(x) = a{(x – h)^2} + k\). Write down the value of \(k\).

[1]
d(iii).

The graph of \(f\) has its vertex on the \(x\)-axis.

The graph of a function \(g\) is obtained from the graph of \(f\) by a reflection of \(f\) in the \(x\)-axis, followed by a translation by the vector \(\left( \begin{array}{c}0\\6\end{array} \right)\). Find \(g\), giving your answer in the form \(g(x) = A{x^2} + Bx + C\).

[4]
e.
Answer/Explanation

Markscheme

correct value \(0\), or \(36 – 12p\)     A2     N2

[2 marks]

a(i).

correct equation which clearly leads to \(p = 3\)     A1

eg     \(36 – 12p = 0,{\text{ }}36 = 12p\)

\(p = 3\)     AG     N0

[1 mark]

a(ii).

METHOD 1

valid approach     (M1)

eg     \(x =  – \frac{b}{{2a}}\)

correct working     A1

eg     \( – \frac{{( – 6)}}{{2(3)}},{\text{ }}x = \frac{6}{6}\)

correct answers     A1A1     N2

eg     \(x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)\)

METHOD 2

valid approach     (M1)

eg     \(f(x) = 0\), factorisation, completing the square

correct working     A1

eg     \({x^2} – 2x + 1 = 0,{\text{ }}(3x – 3)(x – 1),{\text{ }}f(x) = 3{(x – 1)^2}\)

correct answers     A1A1     N2

eg     \(x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)\)

METHOD 3

valid approach using derivative     (M1)

eg     \(f'(x) = 0,{\text{ }}6x – 6\)

correct equation     A1

eg     \(6x – 6 = 0\)

correct answers     A1A1     N2

eg     \(x = 1,{\text{ }}y = 0;{\text{ }}(1,{\text{ }}0)\)

[4 marks]

b.

\(x = 1\)     A1     N1

[1 mark]

c.

\(a = 3\)     A1     N1

[1 mark]

d(i).

\(h = 1\)     A1     N1

[1 mark]

d(ii).

\(k = 0\)     A1     N1

[1 mark]

d(iii).

attempt to apply vertical reflection     (M1)

eg     \( – f(x),{\text{ }} – 3{(x – 1)^2}\), sketch

attempt to apply vertical shift 6 units up     (M1)

eg     \( – f(x) + 6\), vertex \((1, 6)\)

transformations performed correctly (in correct order)     (A1)

eg     \( – 3{(x – 1)^2} + 6,{\text{ }} – 3{x^2} + 6x – 3 + 6\)

\(g(x) =  – 3{x^2} + 6x + 3\)     A1     N3

[4 marks]

e.

Question

The following diagram shows the graph of a function \(f\).

Find \({f^{ – 1}}( – 1)\).

[2]
a.

Find \((f \circ f)( – 1)\).

[3]
b.

On the same diagram, sketch the graph of \(y = f( – x)\).

[2]
c.
Answer/Explanation

Markscheme

valid approach     (M1)

eg\(\;\;\;\)horizontal line on graph at \( – 1,{\text{ }}f(a) =  – 1,{\text{ }}( – 1,5)\)

\({f^{ – 1}}( – 1) = 5\)     A1     N2

[2 marks]

a.

attempt to find \(f( – 1)\)     (M1)

eg\(\;\;\;\)line on graph

\(f( – 1) = 2\)     (A1)

\((f \circ f)( – 1) = 1\)     A1     N3

[3 marks]

b.

     A1A1     N2

Note:     The shape must be an approximately correct shape (concave down and increasing). Only if the shape is approximately correct, award the following for points in circles:

A1 for the \(y\)-intercept,

A1 for any two of these points \(( – 5,{\text{ }} – 1),{\text{ }}( – 2,{\text{ }}1),{\text{ }}(1,{\text{ }}2)\).

[2 marks]

Total [7 marks]

c.

Question

The following diagram shows part of the graph of a quadratic function \(f\).

The vertex is at \((1,{\text{ }} – 9)\), and the graph crosses the yaxis at the point \((0,{\text{ }}c)\).

The function can be written in the form \(f(x) = {(x – h)^2} + k\).

Write down the value of \(h\) and of \(k\).

[2]
a.

Find the value of \(c\).

[2]
b.

Let \(g(x) =  – {(x – 3)^2} + 1\). The graph of \(g\) is obtained by a reflection of the graph of \(f\) in the \(x\)-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\).

Find the value of \(p\) and of \(q\).

[5]
c.

Find the x-coordinates of the points of intersection of the graphs of \(f\) and \(g\).

[7]
d.
Answer/Explanation

Markscheme

\(h = 1,{\text{ }}k =  – 9\;\;\;\left( {{\text{accept }}{{(x – 1)}^2} – 9} \right)\)     A1A1     N2

[2 marks]

a.

METHOD 1

attempt to substitute \(x = 0\) into their quadratic function     (M1)

eg\(\;\;\;f(0),{\text{ }}{(0 – 1)^2} – 9\)

\(c =  – 8\)     A1     N2

METHOD 2

attempt to expand their quadratic function     (M1)

eg\(\;\;\;{x^2} – 2x + 1 – 9,{\text{ }}{x^2} – 2x – 8\)

\(c =  – 8\)     A1     N2

[2 marks]

b.

evidence of correct reflection     A1

eg\(\;\;\; – \left( {{{(x – 1)}^2} – 9} \right)\), vertex at \((1,{\text{ }}9)\), y-intercept at \((0,{\text{ }}8)\)

valid attempt to find horizontal shift     (M1)

eg\(\;\;\;1 + p = 3,{\text{ }}1 \to 3\)

\(p = 2\)     A1     N2

valid attempt to find vertical shift     (M1)

eg\(\;\;\;9 + q = 1,{\text{ }}9 \to 1,{\text{ }} – 9 + q = 1\)

\(q =  – 8\)     A1     N2

Notes:     An error in finding the reflection may still allow the correct values of \(p\) and \(q\) to be found, as the error may not affect subsequent working. In this case, award A0 for the reflection, M1A1 for \(p = 2\), and M1A1 for \(q =  – 8\).

If no working shown, award N0 for \(q = 10\).

[5 marks]

c.

valid approach (check FT from (a))     M1

eg\(\;\;\;f(x) = g(x),{\text{ }}{(x – 1)^2} – 9 =  – {(x – 3)^2} + 1\)

correct expansion of both binomials     (A1)

eg\(\;\;\;{x^2} – 2x + 1,{\text{ }}{x^2} – 6x + 9\)

correct working     (A1)

eg\(\;\;\;{x^2} – 2x – 8 =  – {x^2} + 6x – 8\)

correct equation     (A1)

eg\(\;\;\;2{x^2} – 8x = 0,{\text{ }}2{x^2} = 8x\)

correct working     (A1)

eg\(\;\;\;2x(x – 4) = 0\)

\(x = 0,{\text{ }}x = 4\)     A1A1     N3

[7 marks]

Total [16 marks]

d.

Question

The following diagram shows the graph of a function \(f\), for −4 ≤ x ≤ 2.

On the same axes, sketch the graph of \(f\left( { – x} \right)\).

[2]
a.

Another function, \(g\), can be written in the form \(g\left( x \right) = a \times f\left( {x + b} \right)\). The following diagram shows the graph of \(g\).

Write down the value of a and of b.

[4]
b.
Answer/Explanation

Markscheme

A2 N2
[2 marks]

a.

recognizing horizontal shift/translation of 1 unit      (M1)

eg  = 1, moved 1 right

recognizing vertical stretch/dilation with scale factor 2      (M1)

eg   a = 2,  ×(−2)

a = −2,  b = −1     A1A1 N2N2

[4 marks]

b.
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