IB DP Maths Topic 2.4 The quadratic function x↦ax2+bx+c : SL Paper 2

 

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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 \(f(x) = 3{x^2}\) . The graph of f is translated 1 unit to the right and 2 units down. The graph of g is the image of the graph of f after this translation.

Write down the coordinates of the vertex of the graph of g .

[2]
a.

Express g in the form \(g(x) = 3{(x – p)^2} + q\) .

[2]
b.

The graph of h is the reflection of the graph of g in the x-axis.

Write down the coordinates of the vertex of the graph of h .

[2]
c.
Answer/Explanation

Markscheme

\((1{\text{, }} – 2)\)     A1A1     N2

[2 marks]

a.

\(g(x) = 3{(x – 1)^2} – 2\) (accept \(p = 1\) , \(q = – 2\) )     A1A1     N2

[2 marks]

b.

\((1{\text{, }}2)\)     A1A1     N2

[2 marks]

c.

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

Let \(f(x) = (x – 1)(x – 4)\).

Find the \(x\)-intercepts of the graph of \(f\).

[3]
a.

The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the \(x\)-axis.

Find the volume of the solid formed.

[3]
b.
Answer/Explanation

Markscheme

valid approach     (M1)

eg     \(f(x) = 0\), sketch of parabola showing two \(x\)-intercepts

\(x = 1,{\text{ }}x = 4{\text{   }}\left( {{\text{accept (1, 0), (4, 0)}}} \right)\)     A1A1     N3

[3 marks]

a.

attempt to substitute either limits or the function into formula involving \({f^2}\)     (M1)

eg     \(\int_1^4 {{{\left( {f(x)} \right)}^2}{\text{d}}x,{\text{ }}\pi \int {{{\left( {(x – 1)(x – 4)} \right)}^2}} } \)

\({\text{volume}} = 8.1\pi {\text{   (exact), 25.4}}\)     A2     N3

[3 marks] 

b.

Question

Let \(f(x) = 5 – {x^2}\). Part of the graph of \(f\)is shown in the following diagram.

The graph crosses the \(x\)-axis at the points \(\rm{A}\) and \(\rm{B}\).

Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).

[3]
a.

The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the \(x\)-axis.

Find the volume of the solid formed.

[3]
b.
Answer/Explanation

Markscheme

recognizing \(f(x) = 0\)     (M1)

eg     \(f = 0,{\text{ }}{x^2} = 5\)

\(x =  \pm 2.23606\)

\(x =  \pm \sqrt 5 {\text{ (exact), }}x =  \pm 2.24\)     A1A1     N3

[3 marks]

a.

attempt to substitute either limits or the function into formula

involving \({f^2}\)     (M1)

eg     \(\pi \int {{{\left( {5 – {x^2}} \right)}^2}{\text{d}}x,{\text{ }}\pi \int_{ – 2.24}^{2.24} {\left( {{x^4} – 10{x^2} + 25} \right)} ,{\text{ }}2\pi \int_0^{\sqrt 5 } {{f^2}} } \)

\(187.328\)

volume \(= 187\)     A2     N3

[3 marks]

b.
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