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Functions, MET1 2021 VCAA 9

Consider the unit circle    and the tangent to the circle at the point , shown in the diagram below.
  

  1. Show that the equation of the line that passes through the points and is given by  .   (2 marks)

Let  ,  where  , and let the graph of the function be the transformation of the line that passes through the points and under .

    1. Find the values of for which the graph of intersects with the unit circle at least once.   (1 mark)

    2. Let the graph of intersect the unit circle twice.
    3. Find the values of for which the coordinates of the points of intersection have only positive values.   (1 mark)

  2. For  , let  be the point of intersection of the graph of with the unit circle, where  is always the point of intersection that is closest to , as shown in the diagram below.
     
         

Let be the function that gives the area of triangle in terms of .

    1. Define the function .   (2 marks)

    2. Determine the maximum possible area of triangle .   (2 marks)

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  2.  i.
  3. ii.
  4. i.
  5. ii. ²
Show Worked Solution

a.   

♦♦♦ Mean mark part (a) 18%.

 
 

 

♦♦♦ Mean mark part (b)(i) 6%.
bi. 

 

b.ii.

 

♦♦♦ Mean mark part (b)(ii) 4%.


 

c.i.   `text(If)\ \ q=1, \ P^{′} = P and theta=pi/3`

♦♦♦ Mean mark part (c)(i) 10%.
 
   

 
c.ii. 

♦♦♦ Mean mark part (c)(ii) 12%.

²

Algebra, MET2 2017 VCAA 4

Let  . Part of the graph of  is shown below.
 

  1. The transformation    maps the graph of    onto the graph of  .

     

    State the values of and .   (2 marks)

  2. Find the rule and domain for  , the inverse function of  .   (2 marks)

  3. Find the area bounded by the graphs of  and  .   (3 marks)

  4. Part of the graphs of  and  are shown below.
     

         
     
    Find the gradient of  and the gradient of    at  .   (2 marks)

The functions of  , where  , are defined with domain such that  .

  1. Find the value of such that  .  (1 mark)

  2. Find the rule for the inverse functions  of  , where  .   (1 mark)

  3. i. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

    ii. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

  4. The lines and are the tangents at the origin to the graphs of    and    respectively.
  5. Find the value(s) of for which the angle between and is 30°.   (2 marks)

  6. Let be the value of for which    has only one solution.
  7.  i. Find .   (2 marks)

  8. ii. Let    be the area bounded by the graphs of    and    for all  .
  9.     State the smallest value of such that  .   (1 mark)

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a. 

b. 

c.  ²

d.

e. 

f. 

g.i. 

g.ii. 

h. 

i.i. 

i.ii. 

Show Worked Solution

a.   

   

 

 

b.   

 

 

c.   

MARKER’S COMMENT: Specifically recommends using    in this type of integral to avoid errors in stating the integral.

 

  ²

 

d.   
 

 

e.   

 

f.   

 

♦♦ Mean mark part (g)(i) 31%.

g.i.   
 

 

 

♦♦ Mean mark part (g)(ii) 30%.

g.ii.   
 

 

 

h.   

♦♦♦ Mean mark part (h) 13%.

 

 

 

i.i   

♦♦♦ Mean mark part (i)(i) 4%.

 

 

i.ii 

♦♦♦ Mean mark part (i)(ii) 2%.

 

Graphs, MET2 2013 VCAA 1

Trigg the gardener is working in a temperature-controlled greenhouse. During a particular 24-hour time interval, the temperature   is given by  , where is the time in hours from the beginning of the 24-hour time interval.

  1. State the maximum temperature in the greenhouse and the values of when this occurs.   (2 marks)

  2. State the period of the function    (1 mark)

  3. Find the smallest value of for which     (2 marks)

  4. For how many hours during the 24-hour time interval is     (2 marks)

Trigg is designing a garden that is to be built on flat ground. In his initial plans, he draws the graph of    for    and decides that the garden beds will have the shape of the shaded regions shown in the diagram below. He includes a garden path, which is shown as line segment

  1. The line through points    and    is a tangent to the graph of    at point

    1. Find    when     (1 mark)

    2. Show that the value of is     (1 mark)

In further planning for the garden, Trigg uses a transformation of the plane defined as a dilation of factor from the -axis and a dilation of factor from the -axis, where and are positive real numbers.

  1. Let and be the image, under this transformation, of the points and respectively. 

     

    1. Find the values of and  if    and     (2 marks)

    2. Find the coordinates of the point    (1 mark)

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  5.  i. 
    ii. 
  6.  i. 
    ii. 
Show Worked Solution

a.  

 

b.   
   

 

c.  

 

 

d.  

met2-2013-vcaa-sec1-answer1

 

 

e.i.  

 

e.ii. 

 

 

 

f.i.  

 
♦♦♦ Mean mark part (f)(i) 14%.

 

 
♦♦♦ Mean mark part (f)(ii) 12%.

 

f.ii.   
   

Graphs, MET2 2015 VCAA 11 MC

The transformation that maps the graph of    onto the graph of    is a

  1. dilation by a factor of from the -axis.
  2. dilation by a factor of from the -axis.
  3. dilation by a factor of from the -axis.
  4. dilation by a factor of from the -axis.
  5. dilation by a factor of from the -axis.
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Show Worked Solution
♦♦ Mean mark 24%.