IBMaths: AI HLDPExam Questions2. Functions2.2 Further Functions & Graphs

Further Functions & Graphs (DP IB Maths: AI HL)

Exam Questions

4 hours32 questions
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Hard
Very Hard
1a
Sme Calculator1 mark

A function is defined by f open parentheses x close parentheses space equals space 54 x minus 13 comma space space minus 2 less than x less than 20.

Find the value of f open parentheses begin inline style 5 over 2 end style close parentheses.

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1b
Sme Calculator2 marks

Write down the range of f open parentheses x close parentheses.

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1c
Sme Calculator2 marks

Find the value of f to the power of negative 1 end exponent open parentheses 122 close parentheses.

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1d
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Write down the range of the inverse function.

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2a
Sme Calculator2 marks

Consider the function f open parentheses x close parentheses space equals space minus 6 x minus 3. The domain of f open parentheses x close parentheses is negative 5 less or equal than x less or equal than 3.

Find

(i)
f open parentheses 2 close parentheses

(ii)
x when f open parentheses x close parentheses space equals space 15.

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2b
Sme Calculator3 marks

Find the range of f open parentheses x close parentheses.

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2c
Sme Calculator1 mark

Write down the domain of the inverse function.

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3a
Sme Calculator3 marks

Consider the function g open parentheses x close parentheses space equals square root of 4 minus x end root.

Sketch the graph of the function g open parentheses x close parentheses, labelling the x and y intercepts.

q4-2-2-easy-ib-ai-sl-maths

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3b
Sme Calculator2 marks

Find

(i)
g open parentheses negative 5 close parentheses

(ii)
x when g open parentheses x close parentheses space equals space 1 half.

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3c
Sme Calculator2 marks

Find

(i)
the maximum possible domain of the function g open parentheses x close parentheses

(ii)
the range of the function g open parentheses x close parentheses that corresponds to the domain found in part (c) (i).

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4a
Sme Calculator2 marks

Consider the functions f open parentheses x close parentheses space equals space minus x to the power of 5 space plus 2020 and g open parentheses x close parentheses space equals space 1 over square root of open parentheses 1 minus x close parentheses end root cubed minus 2.

Find the coordinates of the y-intercepts for the graph of

(i)
f

(ii)
g.

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4b
Sme Calculator2 marks

Find the coordinates of the x-intercepts for the graph of

(i)
f

(ii)
g.

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4c
Sme Calculator3 marks

For the graph of g, find the equation of

(i)
the vertical asymptote

(ii)
the horizontal asymptote.

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5a
Sme Calculator2 marks

Consider the functions f open parentheses x close parentheses space equals space x to the power of negative 4 end exponent minus 2021 and g left parenthesis x right parenthesis equals 2 minus square root of x minus 1 end root.
Find the maximum possible domain and range of g.

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5b
Sme Calculator3 marks

For the graph of f, find the equation of

(i)
the vertical asymptote

(ii)
the horizontal asymptote.

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5c
Sme Calculator2 marks

Find the coordinates of the x-intercepts for the graph of

(i)
f

(ii)
g.

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6a
Sme Calculator4 marks

Consider the functions f left parenthesis x right parenthesis space equals space minus x squared minus x plus 6 and g left parenthesis x right parenthesis equals left parenthesis 2 x plus 1 right parenthesis squared minus 9.

Sketch the graphs of the functions f open parentheses x close parentheses and g open parentheses x close parentheses and label the coordinates of the vertices for both functions.

q6a-2-2-easy-ib-ai-sl-maths

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6b
Sme Calculator2 marks

Find the coordinates for the points of intersection of f open parentheses x close parentheses and g open parentheses x close parentheses.

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6c
Sme Calculator2 marks

Find the x-intercepts of f open parentheses x close parentheses and g open parentheses x close parentheses.

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7a
Sme Calculator1 mark

The diagram below shows part of the graph of the function f left parenthesis x right parenthesis equals negative x squared plus b x plus c, where b and c are both integers. Points P(-2, 0) and R(6, 0) represent the x-intercepts, point Q(0, 12) represents the straight y-intercept, point V represents the vertex of the graph of f and O represents the origin (0, 0).

4s4VVZsH_q7a-2-2-medium-ib-ai-sl-maths

Write down the value of c.

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7b
Sme Calculator3 marks

Find the value of b and write down f left parenthesis x right parenthesis.

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7c
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Write down the coordinates of straight V.

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8a
Sme Calculator4 marks

The function g open parentheses x close parentheses space equals space a x squared plus b x plus c intercepts the y-axis at negative 16, has an x-intercept when x space equals space minus 4 and can be obtained by an appropriate translation of the graph y space equals space 2 x squared.

(i)
Find the values of a comma space b and c.

(ii)
Write down g left parenthesis x right parenthesis

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8b
Sme Calculator1 mark

Find the other x-intercept of g left parenthesis x right parenthesis.

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8c
Sme Calculator2 marks

Write down the coordinates of the vertex of g left parenthesis x right parenthesis.

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9a
Sme Calculator1 mark

The diagram below shows the graph of the function f open parentheses x close parentheses space equals space 2 sin open parentheses 2 x close parentheses for 0 degree less or equal than space x less or equal than 360 degree.

q8a-2-2-medium-ib-ai-sl-maths

State the amplitude of f left parenthesis x right parenthesis.

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9b
Sme Calculator2 marks

Calculate the period of f left parenthesis x right parenthesis.

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9c
Sme Calculator4 marks

Find the possible values of x when f open parentheses x close parentheses space equals space minus 1.

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10a
Sme Calculator4 marks

The diagram below shows part of the graph of the function f left parenthesis x right parenthesis equals x cubed minus x squared minus 4 x plus 1.

q10-2-2-easy-ib-ai-sl-maths

Points A comma space C comma space D and F represent where the graph of f intersects the coordinate axes, write down the coordinates for

(i)
A

(ii)
C

(iii)
D

(iv)
F.

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10b
Sme Calculator2 marks

Points B and E represent the local maximum and minimum respectively for f open parentheses x close parentheses, write down the coordinates for

(i)
B

(ii)
E.

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11a
Sme Calculator2 marks

The diagram below shows part of the graph of the function f left parenthesis x right parenthesis equals 2 to the power of x minus 3.

q11a-2-2-medium-ib-ai-sl-maths

Find

(i)
f left parenthesis 2 right parenthesis

(ii)
x spacewhen f left parenthesis x right parenthesis equals negative 1.

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11b
Sme Calculator1 mark

The point P represents the y-intercept of f left parenthesis x right parenthesis. spaceWrite down the coordinates of P.

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11c
Sme Calculator1 mark

The point Q represents the x-intercept of f left parenthesis x right parenthesis. Write down the coordinates of Q.

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11d
Sme Calculator2 marks

Draw the line y equals negative 3 on the graph above.

Write down the number of solutions to the equation f left parenthesis x right parenthesis equals negative 3.

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1a
Sme Calculator1 mark

Let f left parenthesis x right parenthesis equals x squared minus 3 x plus 2. The diagram below shows part of the graph of f.

q1a-2-2-hard-ib-ai-sl-maths

Another function is defined by g left parenthesis x right parenthesis equals 2 minus x.

Sketch the graph of g on the axes above.

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1b
Sme Calculator3 marks

The graph of f and g intersect at points A and B.

Find the coordinates of A and B and label them on the diagram above.

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1c
Sme Calculator2 marks

Find the length of the line AB.

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2a
Sme Calculator2 marks

Let f left parenthesis x right parenthesis equals fraction numerator 7 over denominator 2 left parenthesis x minus 7 right parenthesis end fraction space minus 5, for x not equal to 7.

For the graph of f, find the:

(i)
x-intercept

(ii)
y-intercept.

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2b
Sme Calculator2 marks

For the graph of f, write down the equation of any asymptotes.

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2c
Sme Calculator2 marks

Let g left parenthesis x right parenthesis equals 2 left parenthesis 1 minus 2 x right parenthesis comma spacefor x element of straight real numbers. The graphs of f and g intersect at points straight P and straight Q.

Write down the coordinates of P and Q.

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2d
Sme Calculator2 marks

Find the distance of PQ.

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3a
Sme Calculator4 marks

The perimeter, P, and area, A, of a given square can be expressed by straight P equals 4 x space and straight A equals x squared respectively, where  is the length of the side of the square.

Write down an expression for:

(i)
P in terms of A, straight P left parenthesis straight A right parenthesis

(ii)
A in terms of P, straight A left parenthesis straight P right parenthesis.

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3b
Sme Calculator2 marks

straight P to the power of negative 1 end exponent left parenthesis 40 right parenthesis equals straight A left parenthesis k right parenthesis.

Find the value of k and straight A left parenthesis k right parenthesis.

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4a
Sme Calculator2 marks

Consider the function f left parenthesis x right parenthesis equals a left parenthesis 0.75 right parenthesis to the power of x plus b where a and b are constants. The graph of f passes through the points left parenthesis 0 comma 18 right parenthesis space and   left parenthesis 2 comma 11 right parenthesis  and is shown below.

l7Mb5_Ct_q4a-2-2-hard-ib-ai-sl-maths

Write down two equations relating a spaceand b.

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4b
Sme Calculator2 marks

Find the value of a spaceand b.

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4c
Sme Calculator2 marks

Write down the equation of the horizontal asymptote of the graph of f.

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5a
Sme Calculator4 marks

The function f left parenthesis x right parenthesis equals a x squared plus b x plus c intercepts the y-axis at negative 12 and has an x-intercept at x equals 3. The function can be obtained by an appropriate shift of the graph y equals negative 4 x squared.

Find the values of a comma space b and c.

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5b
Sme Calculator1 mark

Find the other x-intercept of f left parenthesis x right parenthesis.

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5c
Sme Calculator2 marks

Determine the coordinates of the maximum value of f left parenthesis x right parenthesis .

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6a
Sme Calculator1 mark

A function is defined by f left parenthesis x right parenthesis equals fraction numerator 1 over denominator left parenthesis x minus 3 right parenthesis squared end fraction plus 2 comma space x not equal to p.

Find the value of p.

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6b
Sme Calculator2 marks

For the graph of f write down the equation of any asymptotes.

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6c
Sme Calculator1 mark

Find the range of f.

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6d
Sme Calculator4 marks

The line  intersects the graph of f when x equals 1 and when x equals 4.

Find the equation of l. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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7a
Sme Calculator2 marks

A function is defined by f left parenthesis x right parenthesis equals 4 minus space fraction numerator 12 over denominator left parenthesis 5 x plus 9 right parenthesis end fraction comma space x not equal to a.

Find the value of a. Give your answer as a fraction.

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7b
Sme Calculator3 marks

Find the range of f.

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7c
Sme Calculator2 marks

Find the value of f to the power of negative 1 end exponent left parenthesis 2 right parenthesis. Give your answer as a fraction.

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8a
Sme Calculator2 marks

The diagram below shows the graph of the function f left parenthesis x right parenthesis equals 8 cos space open parentheses 48 x close parentheses minus 3, for negative 5 less or equal than x less or equal than 5.

q6a-2-2-hard-ib-ai-sl-maths

For the graph of f, find the:

(i)
x-intercepts

(ii)
y-intercept.

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8b
Sme Calculator1 mark

Write down the range of f.

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8c
Sme Calculator4 marks

Three lines are drawn connecting the two local minimum points and the local maximum point forming a triangle. Calculate the area of the triangle.

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9a
Sme Calculator4 marks

Let  f left parenthesis x right parenthesis equals 2 minus 5 space cos space open parentheses 30 open parentheses 3 space x minus 1 close parentheses close parentheses comma for x greater than 0.

The nth maximum point on the graph of f has x coordinate x subscript n, where n space element of space straight integer numbers to the power of plus.

Given that x subscript n space equals space x subscript 1 plus open parentheses n minus 1 close parentheses d, find x subscript 1 and d.

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9b
Sme Calculator4 marks

Using sigma notation, write down an expression for x subscript 1 plus x subscript 2 plus x subscript 3 plus midline horizontal ellipsis x subscript 8.

Find the value of the sum from part (b) (i). Give your answer to 2 decimal places.

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10a
Sme Calculator2 marks

The average fat-free mass, M, in kg, of footballers as a function of their age, a, in years, can be given by the logarithmic function:

straight M open parentheses a close parentheses equals 10 log open parentheses a minus 15 close parentheses plus 50 comma space space space space space space space space space space space space space space space space 16 less or equal than a less or equal than 25.

Calculate the average fat free mass of players aged:

(i)
16 years

(ii)
25 years.

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10b
Sme Calculator3 marks

Find an expression for a linear model using your answers to part (a) (i) and (ii).

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10c
Sme Calculator3 marks

Calculate the percentage error from using the linear model found in part (b) to approximate the average fat free mass of a player aged 20 years old.

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11a
Sme Calculator2 marks

The axes below shows the graph of the piecewise function, f

q11a-2-2-hard-ib-ai-sl-maths

The gradient of the graph of f is 0 for a less or equal than x less than b.

Find the values of a and b.

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11b
Sme Calculator4 marks

Complete the following piecewise function for f

f open parentheses x close parentheses space equals space open curly brackets space space space space space space space 0 less or equal than x less than 2 space space space space space space space 2 less or equal than x less than 3 space space space space space space space 3 less or equal than x less or equal than 4 close

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1a
Sme Calculator2 marks

A function is defined by f left parenthesis x right parenthesis equals e to the power of x squared plus b x plus 4 end exponent. The graph of  has an axis of symmetry of x equals 2.

Find the value of b.

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1b
Sme Calculator1 mark

Find the range of f .

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1c
Sme Calculator3 marks

Another function is defined by g open parentheses x close parentheses space equals space minus fraction numerator open parentheses x squared minus 25 close parentheses over denominator 5 end fraction. The graph of f and g intersect at points A and B.

Find the equation of the line passing through points A and B. Give your answer in the form y equals m x plus c.

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1d
Sme Calculator2 marks

Find the distance of the line AB.

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2a
Sme Calculator2 marks

Consider the function f left parenthesis x right parenthesis equals 5 minus log left parenthesis 6 minus 4 x right parenthesis. The line l subscript 1 intersects the graph of f at point Aleft parenthesis negative 1 comma y right parenthesis and Bleft parenthesis x comma 5 right parenthesis.

Find the value of x and y.

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2b
Sme Calculator2 marks

Find the equation of l subscript 1. Give your answer in the form y equals m x plus c, where m and c are fractions.

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2c
Sme Calculator4 marks

The line l subscript 2 is the perpendicular bisector to [AB].

Find the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

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3a
Sme Calculator2 marks

A rectangle has length l equals 4 x and width w equals x.

Find an expression for

(i)
the perimeter of the rectangle, P, in terms of x.

(ii)
the area of the rectangle, A, in terms of x.

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3b
Sme Calculator2 marks

Show that P left parenthesis straight A right parenthesis equals 5 square root of straight A.

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3c
Sme Calculator3 marks

The graph of the function P, for 0 less or equal than A less or equal than 4, is shown below.

q3a-2-2-very-hard-ib-al-

On the grid above, draw the graph of the inverse function P to the power of negative 1 end exponent.

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4a
Sme Calculator4 marks

The function f is a quadratic in the form f left parenthesis x right parenthesis equals a x squared plus b x minus 2, for negative 10 less or equal than x less or equal than 10.

The graph of f has x-intercepts open parentheses fraction numerator 1 plus square root of 5 over denominator 2 end fraction comma space 0 close parentheses and open parentheses fraction numerator 1 minus square root of 5 over denominator 2 end fraction comma space 0 close parentheses.

Find the values of a and b.

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4b
Sme Calculator2 marks

Another function can be defined by g left parenthesis x right parenthesis equals 6 open parentheses 0.8 close parentheses to the power of negative x end exponent minus 1, for negative 10 less or equal than x less or equal than 10.

The graph of f and g intersect at points straight A and straight B.

Find the coordinates of straight A and straight B.

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4c
Sme Calculator2 marks

Solve the inequality f left parenthesis x right parenthesis less than g left parenthesis x right parenthesis.

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5a
Sme Calculator4 marks

Let f left parenthesis x right parenthesis equals e to the power of 2 c o s open parentheses 30 x close parentheses end exponent plus 1 comma for x greater than 0.

The nth minimum point on the graph of f has x coordinate x subscript n, where n space element of space straight integer numbers to the power of plus.

Given that x subscript n equals x subscript 1 plus left parenthesis n minus 1 right parenthesis d, find x subscript 1 and space d.

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5b
Sme Calculator4 marks
(i)
Using sigma notation, write down an expression for x subscript 1 plus x subscript 2 plus x subscript 3 plus midline horizontal ellipsis space plus x subscript 16.

(ii)
Find the value of the sum from part (b) (i). Give your answer to the nearest integer.

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6a
Sme Calculator3 marks

Let f left parenthesis x right parenthesis equals 2 x to the power of 4 minus 2 x cubed minus 4 x squared plus x plus 1 where x space element of space straight real numbers.

Solve the inequality f left parenthesis x right parenthesis less than 0.

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6b
Sme Calculator3 marks

For the graph of f, find the coordinates of the

(i)
local maximum point.

(ii)
local minimum points.

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6c
Sme Calculator3 marks

Write down the possible domains of f for which f has an inverse and explain why the domain must be restricted.

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7a
Sme Calculator1 mark

The graph of a quadratic function has equation y equals 1 fourth x squared plus b x plus c, where b comma space c space element of space straight integer numbers comma and the axis of symmetry is x equals negative 4.

q5a-2-2-very-hard-ib-ai-sl-maths

Draw the axis of symmetry on the grid above.

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7b
Sme Calculator3 marks

The graph of the quadratic function intersects the x-axis at points straight A left parenthesis negative 6 comma 0 right parenthesis and B.

Write down the coordinates of B.

Find the values of b and c.

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7c
Sme Calculator4 marks
(i)
Mark and label A and B on the grid above.

(ii)
Write down the coordinates of the Vertex, V, and label it on the grid above.

(iii)
Write down the coordinates of the y-intercept, C, and label it on the grid above.

(iv)
Draw the graph of the quadratic function on the grid above.

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8a
Sme Calculator4 marks

The diagram below shows part of the graph of the function f left parenthesis x right parenthesis equals 5 space cos left parenthesis 12 x minus 30 right parenthesis plus 6, where points A, B, C and D represent stationary points. The points A, B, C and D are joined together forming a quadrilateral ABDC.

q6a-2-2-very-hard-ib-ai-sl-maths

Write down the coordinates for

(i)
straight A.

(ii)
straight B.

(iii)
straight C.

(iv)
straight D.

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8b
Sme Calculator2 marks

Find the area of the quadrilateral ABDC.

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9a
Sme Calculator2 marks

The intensity of light, I, is assumed to be 100% at the surface of the ocean and decreases with depth, d, and can be estimated by the function

I left parenthesis d right parenthesis equals k left parenthesis 1.08 right parenthesis to the power of negative d end exponent

where I is expressed as a percentage, d is the depth below the surface, in metres, and k is a constant.

Calculate the value of k.

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9b
Sme Calculator2 marks

State the domain and range of I.

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9c
Sme Calculator4 marks

On a particularly bright day the intensity of light 6.2 m below the surface is 75%.

Calculate the percentage error between the estimate made by the function and the actual intensity of light.

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10a
Sme Calculator4 marks

Graph the following piecewise function on the grid below

f open parentheses x close parentheses space equals space open curly brackets negative x plus 5 comma space space space space space space space minus 2 less or equal than x less than 2 minus 4 x plus 11 comma   2 less or equal than x less than 3 2 x minus 7 comma   space space space space space space 3 less or equal than x less or equal than 6 close

q10a-2-2-very-hard-ib-ai-sl-maths

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10b
Sme Calculator3 marks

Find the values of x when f open parentheses x close parentheses space equals space 0.

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