IBMaths: AA HLDPExam Questions2. Functions2.9 Further Functions & Graphs

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

Exam Questions

4 hours27 questions
Medium
Hard
Very Hard
1a
Sme Calculator3 marks
a)
Sketch the graph of y equals left parenthesis x minus 1 right parenthesis squared minus 2 vertical line x minus 1 vertical line minus 1 comma for negative 3 less or equal than x less or equal than 6.

How did you do?

1b
Sme Calculator2 marks
b)
Hence, solve the equationspace y equals left parenthesis x minus 1 right parenthesis squared minus 2 vertical line x minus 1 vertical line minus 1 equals 0.

How did you do?

Did this page help you?

2
Sme Calculator7 marks

Given that  

space f left parenthesis x right parenthesis equals ln space x comma space space space space space space space space space x greater than 0 

sketch on separate axes the graphs of 

(i)
y equals f left parenthesis x right parenthesis space

(ii)
y equals vertical line f left parenthesis x right parenthesis vertical line

(iii)
y equals negative f left parenthesis x minus 3 right parenthesis 

On each diagram, show the x-intercepts along with any asymptotes, including their equations.

How did you do?

Did this page help you?

3a
Sme Calculator3 marks

The graph of y equals f left parenthesis x right parenthesis is given below.

q3a_2-9_medium_ib-aa-hl-mathsOn separate axes, draw the graphs of 

a)
vertical line f left parenthesis x right parenthesis vertical line

How did you do?

3b
Sme Calculator3 marks
b)
stretchy left square bracket f stretchy left parenthesis x stretchy right parenthesis stretchy right square bracket squared

How did you do?

Did this page help you?

4a
Sme Calculator3 marks
a)
Sketch the curve begin mathsize 16px style y equals fraction numerator 3 over denominator x plus 4 end fraction end style and line y equals 4 minus x on the same axes, clearly indicating any x- and  y- intercepts and any asymptotes.

How did you do?

4b
Sme Calculator5 marks
b)
Consider the equation 

4 minus x equals open vertical bar fraction numerator 3 over denominator x plus 4 end fraction close vertical bar 

(i)
Explain why the cases x less than negative 4 comma x equals negative 4 and x greater than negative 4 must be considered separately in attempting to solve the equation. 

(ii)
Hence find the exact solutions to the equation.

How did you do?

Did this page help you?

5a
Sme Calculator3 marks

Consider the function space f defined byspace f left parenthesis x right parenthesis equals 3 x squared space arcsin space x, negative 1 less or equal than x less or equal than 1.

a)
Sketch the graph of y equals space f left parenthesis x right parenthesis.

How did you do?

5b
Sme Calculator2 marks
b)
State the range ofspace f.

How did you do?

5c
Sme Calculator3 marks
c)
Solve the inequality  vertical line 3 x squared arcsin space x vertical line greater than 1.

How did you do?

Did this page help you?

6a
Sme Calculator4 marks

Consider the functionspace f defined byspace f left parenthesis x right parenthesis equals square root of 9 minus x end root, wherespace f has the largest possible valid domain.

a)
(i)
Sketch the graph of y equals f left parenthesis x right parenthesis, labelling the  x- and  y-intercepts. 

(ii)
State the domain and range ofspace f.

How did you do?

6b
Sme Calculator4 marks
b)
(i)
On the same set of axes, sketch the graph of the functionspace f left parenthesis vertical line x vertical line right parenthesis, labelling the x- and y-intercepts.

(ii)
State the domain and range of the functionspace f left parenthesis vertical line x vertical line right parenthesis.

How did you do?

Did this page help you?

7a
Sme Calculator4 marks

 Let begin mathsize 16px style space f left parenthesis x right parenthesis equals fraction numerator 7 minus 9 x over denominator c x minus 12 end fraction end style, begin mathsize 16px style x not equal to 12 over c end style,  where c is a non-zero constant. 

The line x equals 4 is a vertical asymptote to the graph of y equals f left parenthesis x right parenthesis. 

a)
(i)
Find the value of c.

(ii)
State the equation of the horizontal asymptote to the graph of y equals f left parenthesis x right parenthesis.

How did you do?

7b
Sme Calculator3 marks
b)
The line y equals k comma where k element of straight real numbers comma intersects the graph of y equals vertical line f left parenthesis x right parenthesis vertical line at exactly one point. Find the possible values of k.

How did you do?

Did this page help you?

8a
Sme Calculator3 marks

Let f left parenthesis x right parenthesis equals 2 x cubed minus 2 x , for x element of straight real numbers  .

(a)
(i)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar.
 
(ii)
State the transformation of the graph y equals f left parenthesis x right parenthesis  to y equals open vertical bar f left parenthesis x right parenthesis close vertical bar for f left parenthesis x right parenthesis less than 0.

How did you do?

8b
Sme Calculator3 marks
(b)
(i)
Sketch the graph of y equals f left parenthesis open vertical bar x close vertical bar right parenthesis.

 

          (ii)    State the transformation of the graph y equals f left parenthesis x right parenthesis to  y equals f left parenthesis open vertical bar x close vertical bar right parenthesis  for x less than 0.

How did you do?

Did this page help you?

9a
Sme Calculator3 marks

Let f open parentheses x close parentheses equals x open parentheses x minus 2 close parentheses.

(a)
Sketch the graph of y equals f open parentheses x close parentheseson the coordinate axes below. Be sure to label anywhere the graph intersects the coordinate axes and any extrema.
q9_2-9_ib-aa-hl_further-functions--graphs_medium_dig

How did you do?

9b
Sme Calculator3 marks
(b)
On the same axes, sketch the graph of the reciprocal y equals fraction numerator 1 over denominator f left parenthesis x right parenthesis end fraction.Be sure to label anywhere the graph intersects the coordinate axes and any extrema.

How did you do?

9c
Sme Calculator2 marks
(c)
Find the equation of the horizontal and vertical asymptotes of the graph of y equals f left parenthesis x right parenthesis.

How did you do?

Did this page help you?

1a
Sme Calculator4 marks

Consider the function f left parenthesis x right parenthesis equals x squared minus 4 vertical line x vertical line minus 5 comma space x element of R. 

(a)
Solve f left parenthesis x right parenthesis equals 0.

How did you do?

1b
Sme Calculator4 marks
(b)
Sketch the graph of f. Clearly indicate the intersections with the coordinate axes and any turning points.

How did you do?

Did this page help you?

2
Sme Calculator9 marks

Given that

f left parenthesis x right parenthesis equals e to the power of x comma x element of R comma 

sketch on separate axes the graphs of

(i)
y equals f left parenthesis vertical line x minus 1 vertical line right parenthesis
(ii)
y equals vertical line f left parenthesis x right parenthesis minus 1 vertical line
(iii)
y equals f left parenthesis negative vertical line x vertical line right parenthesis.

Show any intercepts with the axes, label any local maximum and minimum points and give the equations of any asymptotes. Leave numbers in terms of e where appropriate.

How did you do?

Did this page help you?

3a
Sme Calculator6 marks

Consider the function f defined by f left parenthesis x right parenthesis equals 4 x squared minus 8 x minus 5.

(a)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Clearly indicate any intercepts with the axes and any turning points.

 

How did you do?

3b
Sme Calculator3 marks
(b)
Sketch the graph of y equals open square brackets f open parentheses x close parentheses close square brackets squared. Clearly indicate any intercepts with the axes and any turning points.

How did you do?

Did this page help you?

4a
Sme Calculator3 marks
(a)
Sketch the curve y equals fraction numerator 4 over denominator x plus 3 end fraction and the line y equals 3 minus x on the same diagram, clearly indicating any x- and  y- axes intercepts as well as any asymptotes.

How did you do?

4b
Sme Calculator5 marks
(b)
Hence find the exact solutions to the equation
3 minus x equals open vertical bar fraction numerator 4 over denominator x plus 3 end fraction close vertical bar.

How did you do?

Did this page help you?

5
Sme Calculator6 marks

The graph of f has two asymptotes with equation x equals 0.25 and y equals 0.5 as shown below. The graph passes through the points straight A left parenthesis 0 comma 6 right parenthesis and straight B left parenthesis 3 comma space 0 right parenthesis.

q5-2-9-ib-aa-hl-further-functions-_-graphs-hard-dig

Sketch the graph of y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction. Clearly indicate the points where the graph intersects the axes or has a discontinuity and state the equations of any asymptotes.

How did you do?

Did this page help you?

6a
Sme Calculator4 marks

Consider the function  defined by f open parentheses x close parentheses equals ln open parentheses k minus x close parentheses where f has the largest possible valid domain and k is a positive constant such that k greater than 1.

(a)
Sketch the graph of y equals f open parentheses x close parentheses. Give the equations of any asymptotes and any intercepts with the axes in terms of k. Clearly state the domain and range of f.

How did you do?

6b
Sme Calculator4 marks

The function g is defined by g open parentheses x close parentheses equals f open parentheses open vertical bar x close vertical bar close parentheses. The range of g is g open parentheses x close parentheses less or equal than 1.

(b)
(i)
Find the exact value of k.
(ii)
State the domain of g .
(iii)
Sketch the graph of g.

How did you do?

6c
Sme Calculator3 marks
(c)
Given that open vertical bar g open parentheses x close parentheses close vertical bar equals p  has exactly two distinct real solutions find the range of values of p.

How did you do?

Did this page help you?

7
Sme Calculator6 marks

Find the set of values of x which satisfy the inequality

open vertical bar 2 x squared minus 13 x plus 15 close vertical bar greater than 3 x minus 15

How did you do?

Did this page help you?

8a
Sme Calculator2 marks
(a)
By considering the inverse of an appropriate function, sketch the graph y equals square root of x .

How did you do?

8b
Sme Calculator3 marks

The function f is defined by f open parentheses x close parentheses equals square root of 16 minus 2 x end root and its domain is the largest possible set of real values. 

(b)      (i)     State the domain and range of f . 

(ii)    Sketch the graph of y equals f open parentheses x close parentheses. Clearly label the points where the graph intersects the axes.

How did you do?

8c
Sme Calculator3 marks
(c)
On separate sets of axes, sketch the graphs of:
(i)
y equals f open parentheses open vertical bar x close vertical bar close parentheses
(ii)
y equals open square brackets f open parentheses x close parentheses close square brackets squared.
 
For each graph, define the domain and range and clearly label the points where the graph intersects the axes.

How did you do?

Did this page help you?

1a
Sme Calculator5 marks
(a)
By considering graph transformations of an appropriate quadratic function, sketch the graph of y equals open parentheses x plus 4 close parentheses squared minus 2 open vertical bar x plus 4 close vertical bar minus 3 . Clearly indicate any x-intercepts and any
y-intercepts.

 

How did you do?

1b
Sme Calculator4 marks
(b)
Hence, find the values of k such that the equation left parenthesis x plus 4 right parenthesis squared minus 2 vertical line x plus 4 vertical line minus 3 equals k has exactly 4 distinct real solutions.

How did you do?

Did this page help you?

2a
Sme Calculator4 marks

A function f is defined by f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis left parenthesis x minus 3 right parenthesis comma space space x element of R. 

(a)
(i)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar.
(ii)
Hence find the set of values of x for which f open parentheses x close parentheses less than open vertical bar f open parentheses x close parentheses close vertical bar

How did you do?

2b
Sme Calculator5 marks
(b)
Solve the equation f left parenthesis x right parenthesis equals vertical line x squared minus 4 x plus 3 vertical line.

How did you do?

Did this page help you?

3a
Sme Calculator6 marks

A function f is defined by f left parenthesis x right parenthesis equals x squared minus 8 x plus 15 comma space x element of straight real numbers.

(a)
Sketch the graph of y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction.  Clearly indicate any intercepts with the coordinate axes and state the equations of any asymptotes. Find the coordinates of any turning points.

How did you do?

3b
Sme Calculator6 marks
(b)
By sketching the graphs of  y equals f open parentheses x close parentheses and y on the same axes, find the values of x  for which  f open parentheses x close parentheses less or equal than fraction numerator 1 over denominator f open parentheses x close parentheses end fraction.

How did you do?

Did this page help you?

4a
Sme Calculator4 marks

Consider the functionf open parentheses x close parentheses equals fraction numerator open parentheses x plus 3 close parentheses open parentheses x minus 4 close parentheses over denominator x minus 5 end fraction comma space x element of straight real numbers comma space x not equal to 5. spaceThe graph of f  is shown below.

q4a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

(a)
Find the equation of the oblique asymptote of the graph of f .

How did you do?

4b
Sme Calculator5 marks
(b)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

How did you do?

4c
Sme Calculator4 marks
(c)
Sketch the graph of y equals f open parentheses open vertical bar x close vertical bar close parentheses. Clearly indicate the points where the graph crosses the coordinates axes and state the equations of the asymptotes.

How did you do?

Did this page help you?

5a
Sme Calculator4 marks

The graph of a function f is shown below. The equations of the asymptotes are x equals 0, y=-1 and y equals 1.The point A(1, 2)  lies on the graph.

q5a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

On separate sets of axes, sketch the graphs defined below. For each sketch, clearly label any asymptotes or discontinuities and clearly show the coordinates of the point where A gets mapped to.

(a)
y equals fraction numerator 1 over denominator f open parentheses x close parentheses end fraction plus 2.

How did you do?

5b
Sme Calculator4 marks
(b)
y equals open square brackets f open parentheses x close parentheses minus 2 close square brackets squared.

How did you do?

5c
Sme Calculator4 marks
(c)
y equals 1 over open square brackets f open parentheses x close parentheses close square brackets squared

How did you do?

Did this page help you?

6a
Sme Calculator4 marks

Let f be the function defined by f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x squared plus x minus 26 right parenthesis. 

(a)
Sketch the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar. Give the exact coordinates of the x-intercepts.

How did you do?

6b
Sme Calculator6 marks
(b)
Use differentiation to find the set of values of k such that there are 4 points of intersection between the graph of y equals open vertical bar f open parentheses x close parentheses close vertical bar and the line y equals k.

How did you do?

Did this page help you?

7a
Sme Calculator4 marks

The graph of the function f left parenthesis x right parenthesis equals a vertical line x plus p vertical line plus q space spaceis shown below, where a comma space p space comma space q element of straight real numbers. The graph has a local maximum at the point A(3, 5)  and intersects the y-axis at (0,-7).

q7a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

 

(a)
Find the values of a comma space p and q.Hence solve f open parentheses x close parentheses equals 0.

How did you do?

7b
Sme Calculator4 marks
(b)
Find the solutions to the equation f left parenthesis x right parenthesis equals open vertical bar 7 minus 2 x close vertical bar.

How did you do?

Did this page help you?

8a
Sme Calculator2 marks

Consider the function defined by f left parenthesis x right parenthesis equals x vertical line x vertical line comma x element of R.

(a)
Sketch the graph of f.

How did you do?

8b
Sme Calculator3 marks
(b)
By first sketching the graph of f to the power of negative 1 end exponent  on the same set of axes as the graph of f, solve f left parenthesis x right parenthesis equals f to the power of negative 1 end exponent left parenthesis x right parenthesis.

How did you do?

8c
Sme Calculator3 marks
(c)
Given that k is a real constant, find an expression for f to the power of negative 1 end exponent open parentheses k close parentheses in the case when:
(i)
k greater or equal than 0 comma
(ii)
k less than 0.

How did you do?

Did this page help you?

9a
Sme Calculator2 marks

The graph of a function f is shown below.

q9a-2-9-ib-aa-hl-further-functions-_-graphs-very-hard-dig

On separate sets of axes sketch the following graphs clearly showing any key points.

a)
y equals vertical line f left parenthesis x right parenthesis vertical line minus f left parenthesis x right parenthesis.

How did you do?

9b
Sme Calculator4 marks
(b)
y equals f left parenthesis x right parenthesis vertical line f left parenthesis x right parenthesis vertical line

How did you do?

9c
Sme Calculator2 marks
(c)
y equals fraction numerator f open parentheses x close parentheses over denominator open vertical bar f open parentheses x close parentheses close vertical bar end fraction

How did you do?

Did this page help you?

10
Sme Calculator4 marks

 f left parenthesis x right parenthesis equals x to the power of 4 minus 8 x cubed minus 40 x squared plus 224 x minus 240.

Find the values of  such that open vertical bar f open parentheses open vertical bar x close vertical bar close parentheses close vertical bar equals k has six distinct, real solutions.
Explain each stage of your solution in full.

How did you do?

Did this page help you?