IBDP Maths AA: Topic: SL 2.6:The quadratic function: IB style Questions SL Paper 2

Question

Let \(f(x) = 2{x^2} + 4x – 6\) .

Express \(f(x)\) in the form \(f(x) = 2{(x – h)^2} + k\) .[3]

a.

Write down the equation of the axis of symmetry of the graph of f .[1]

b.

Express \(f(x)\) in the form \(f(x) = 2(x – p)(x – q)\) .[2]

c.
Answer/Explanation

Markscheme

evidence of obtaining the vertex     (M1)

e.g. a graph, \(x = – \frac{b}{{2a}}\) , completing the square

\(f(x) = 2{(x + 1)^2} – 8\)     A2     N3

[3 marks]

a.

\(x = – 1\) (equation must be seen)     A1     N1

[1 mark]

b.

\(f(x) = 2(x – 1)(x + 3)\)    A1A1     N2

[2 marks]

c.

Question

Let \(f(x) = {x^3} – 4x + 1\) .

Expand \({(x + h)^3}\) .

[2]
a.

Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) – f(x)}}{h}\) to show that the derivative of \(f(x)\) is \(3{x^2} – 4\) .

[4]
b.

The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} – 2)\) is parallel to the tangent at a point Q. Find the coordinates of Q.

[4]
c.

The graph of f is decreasing for \(p < x < q\) . Find the value of p and of q.

[3]
d.

Write down the range of values for the gradient of \(f\) .

[2]
e.
Answer/Explanation

Markscheme

attempt to expand     (M1)

\({(x + h)^3} = {x^3} + 3{x^2}h + 3x{h^2} + {h^3}\)     A1     N2

[2 marks]

a.

evidence of substituting \(x + h\)     (M1)

correct substitution     A1

e.g. \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^3} – 4(x + h) + 1 – ({x^3} – 4x + 1)}}{h}\)

simplifying     A1

e.g. \(\frac{{({x^3} + 3{x^2}h + 3x{h^2} + {h^3} – 4x – 4h + 1 – {x^3} + 4x – 1)}}{h}\)

factoring out h     A1

e.g. \(\frac{{h(3{x^2} + 3xh + {h^2} – 4)}}{h}\)

\(f'(x) = 3{x^2} – 4\)     AG     N0

[4 marks]

b.

\(f'(1) = – 1\)    (A1)

setting up an appropriate equation     M1

e.g. \(3{x^2} – 4 = – 1\)

at Q, \(x = – 1,y = 4\) (Q is \(( – 1{\text{, }}4)\))    A1    A1

[4 marks]

c.

recognizing that f is decreasing when \(f'(x) < 0\)     R1

correct values for p and q (but do not accept \(p = 1.15{\text{, }}q = – 1.15\) )     A1A1     N1N1

e.g. \(p = – 1.15{\text{, }}q = 1.15\) ; \( \pm \frac{2}{{\sqrt 3 }}\) ; an interval such as \( – 1.15 \le x \le 1.15\)

[3 marks]

d.

\(f'(x) \ge – 4\) , \(y \ge – 4\) , \(\left[ { – 4,\infty } \right[\)     A2     N2

[2 marks]

e.

Question

Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) =  – 0.5{x^2} + 5x – 8\) for \(0 \le x \le 9\) .

On the same diagram, sketch the graphs of f and g .

[3]
a.

Consider the graph of \(f\) . Write down

(i)     the x-intercept that lies between \(x = 0\) and \(x = 3\) ;

(ii)    the period;

(iii)   the amplitude.

[4]
b.

Consider the graph of g . Write down

(i)     the two x-intercepts;

(ii)    the equation of the axis of symmetry.

[3]
c.

Let R be the region enclosed by the graphs of f and g . Find the area of R.

[5]
d.
Answer/Explanation

Markscheme

     A1A1A1     N3

Note: Award A1 for f being of sinusoidal shape, with 2 maxima and one minimum, A1 for g being a parabola opening down, A1 for two intersection points in approximately correct position.

[3 marks]

a.

(i)  \((2{\text{, }}0)\) (accept \(x = 2\) )     A1     N1

(ii) \({\text{period}} = 8\)     A2     N2

(iii) \({\text{amplitude}} = 5\)     A1     N1

[4 marks]

b.

(i) \((2{\text{, }}0)\) , \((8{\text{, }}0)\) (accept \(x = 2\) , \(x = 8\) )     A1A1     N1N1

(ii) \(x = 5\) (must be an equation)     A1     N1

[3 marks]

c.

METHOD 1

intersect when \(x = 2\) and \(x = 6.79\) (may be seen as limits of integration)     A1A1

evidence of approach     (M1)

e.g. \(\int {g – f} \) , \(\int {f(x){\rm{d}}x – \int {g(x){\rm{d}}x}}\) , \(\int_2^{6.79} {\left( {( – 0.5{x^2} + 5x – 8) – \left( {5\cos \frac{\pi }{4}x} \right)} \right)}\)

\({\text{area}} = 27.6\)     A2     N3

METHOD 2

intersect when \(x = 2\) and \(x = 6.79\) (seen anywhere)     A1A1

evidence of approach using a sketch of g and f , or \(g – f\) .     (M1)

e.g. area = \(A + B – C\) , \(12.7324 + 16.0938 – 1.18129 \ldots \)

\({\text{area}} = 27.6\)     A2     N3

[5 marks]

d.

Question

Let \(f(x) = 2{x^2} – 8x – 9\) .

(i)     Write down the coordinates of the vertex.

(ii)    Hence or otherwise, express the function in the form \(f(x) = 2{(x – h)^2} + k\) .

[4]
a(i) and (ii).

Solve the equation \(f(x) = 0\) .

[3]
b.
Answer/Explanation

Markscheme

(i) \((2{\text{, }} – 17)\) or \(x = 2\) , \(y = – 17\)     A1A1     N2

(ii) evidence of valid approach     (M1)

e.g. graph, completing the square, equating coefficients

\(f(x) = 2{(x – 2)^2} – 17\)     A1     N2

[4 marks]

a(i) and (ii).

evidence of valid approach     (M1)

e.g. graph, quadratic formula

\( – 0.9154759 \ldots \) , \(4.915475 \ldots \)

\(x = – 0.915\) , \(4.92\)     A1A1     N3

[3 marks]

b.

Question

[Maximum mark: 4] [without GDC]
The diagram represents the graph of the function \(f : x \mapsto  (x – p)(x – q)\).

(a) Write down the values of \(p\) and \(q\).
(b) The function has a minimum value at the point C. Find the x-coordinate of C.

Answer/Explanation

Ans.

(a) \(p=-\frac{1}{2},q=2\) or vice versa

b) By symmetry C is midway between p, q ⇒ x-coordinate is \(\frac{-\frac{1}{2}+2}{2}=\frac{3}{4}\)

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