IBDP Maths AI: Topic: SL 2.5: Quadratic models: IB style Questions SL Paper 1

Question 12 [Maximum mark: 7]

Consider the function  f (x) = ax 2 + bx + c . The graph of y = f (x) is shown in the diagram. The vertex of the graph has coordinates (0.5, −12.5) . The graph intersects the x-axis at two points, (−2, 0) and ( p, 0).

a. Find the value of p . [1]

b. Find the value of

(i)a .

(ii)b .

(iii) c . [5]

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

Answer/Explanation

(a) 3

(b) METHOD 1     

\(0 = 4a -2b+c\),

\(0 = 9a+ 3b+ c\)

\(- \frac{25}{2}= \frac{1}{4}a + \frac{1}{2}\) b + c\)

on solving we get a , b , c

(i) 2 (ii) – 2(iii) -12

METHOD 2   \(−12.5 = a (0.5+ 2)(0.5- 2)\) (i) \(a=2\)

 

\(0 = 2x(3)2+ 3b + c \)

\(0 = 2x(-2)2+ (-2b)+c\)

(ii) b = −2(iii) c = −12

(c) x = 0.5

Question

The following curves are sketches of the graphs of the functions given below, but in a different order. Using your graphic display calculator, match the equations to the curves, writing your answers in the table below.

(the diagrams are not to scale)

Answer/Explanation

Markscheme

(i) B     (A1)

(ii) D     (A1)

(iii) A     (A1)

(iv) E     (A1)

(v) C     (A1)

(vi) F     (A1)     (C6)[6 marks]

Question

Factorise the expression \({x^2} – kx\) .[1]

a.

Hence solve the equation \({x^2} – kx = 0\) .[1]

b.

The diagram below shows the graph of the function \(f(x) = {x^2} – kx\) for a particular value of \(k\).

Write down the value of \(k\) for this function.[1]

c.

The diagram below shows the graph of the function \(f(x) = {x^2} – kx\) for a particular value of \(k\).

Find the minimum value of the function \(y = f(x)\) .[3]

d.
Answer/Explanation

Markscheme

\(x(x – k)\)     (A1)     (C1)[1 mark]

a.

\(x = 0\) or \(x = k\)     (A1)     (C1)

Note: Both correct answers only.[1 mark]

b.

\(k = 3\)     (A1)     (C1)[1 mark]

c.

\({\text{Vertex at }}x = \frac{{ – ( – 3)}}{{2(1)}}\)     (M1)

Note: (M1) for correct substitution in formula.

\(x = 1.5\)     (A1)(ft)

\(y = – 2.25\)     (A1)(ft)

OR

\(f'(x) = 2x – 3\)     (M1)

Note: (M1) for correct differentiation.

\(x = 1.5\)     (A1)(ft)
\(y = – 2.25\)     (A1)(ft)

OR

for finding the midpoint of their 0 and 3     (M1)
\(x = 1.5\)     (A1)(ft)
\(y = – 2.25\)     (A1)(ft)

Note: If final answer is given as \((1.5{\text{, }}{- 2.25})\) award a maximum of (M1)(A1)(A0)[3 marks]

d.

Question

The graph of a quadratic function \(y = f (x)\) is given below.

Write down the equation of the axis of symmetry.[2]

a.

Write down the coordinates of the minimum point.[2]

b.

Write down the range of \(f (x)\).[2]

c.
Answer/Explanation

Markscheme

x = 3     (A1)(A1)     (C2)

Notes: Award (A1) for “ x = ” (A1) for 3.

The mark for x = is not awarded unless a constant is seen on the other side of the equation.[2 marks]

a.

(3, −14)  (Accept x = 3, y = −14)     (A1)(ft)(A1)     (C2)

Note: Award (A1)(A0) for missing coordinate brackets.[2 marks]

b.

y ≥ −14     (A1)(A1)(ft)     (C2)

Notes: Award (A1) for y ≥ , (A1)(ft) for –14.

Accept alternative notation for intervals.[2 marks]

c.

Question

A quadratic curve with equation y = ax (xb) is shown in the following diagram.

The x-intercepts are at (0, 0) and (6, 0), and the vertex V is at (h, 8).

Find the value of h.[2]

a.

Find the equation of the curve.[4]

b.
Answer/Explanation

Markscheme

\(\frac{{0 + 6}}{2} = 3\)   \(h = 3\)     (M1)(A1)     (C2)

Note: Award (M1) for any correct method.[2 marks]

a.

\(y = ax(x – 6)\)     (A1)

\(8 = 3a(- 3)\)     (A1)(ft)

\(a = – \frac{8}{9}\)     (A1)(ft)

\(y = – \frac{8}{9} x (x – 6)\)     (A1)(ft)

Notes: Award (A1) for correct substitution of \(b = 6\) into equation.

Award (A1)(ft) for substitution of their point V into the equation.

OR

\(y = a(x – 3)^2 + 8\)     (A1)(ft)

Note: Award (A1)(ft) for correct substitution of their h into the equation.

\(0 = a(6 – 3)^2 + 8\) OR \(0 = a(0 – 3)^2 + 8\)     (A1)

Note: Award (A1) for correct substitution of an x intercept.

\(a = – \frac{8}{9}\)     (A1)(ft)

\(y = – \frac{8}{9}(x – 3)^2 + 8\)     (A1)(ft)     (C4)[4 marks]

b.
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