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Functions, 2ADV F1 2022 HSC 12

A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.

  1. Find the equation relating `M` and `T`.    (2 marks)

  2. By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}`.   (2 marks)
     

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M &  &  &  \\
\hline \end{array}

 
                   

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a.    `M=180/T`

 b.    

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}       

 

Show Worked Solution
a.    `M` `prop 1/T`
  `M` `=k/T`
  `12` `=k/15`
  `k` `=15 xx 12`
    `=180`

 
`:.M=180/T`
 


♦ Mean mark (a) 49%.

b.   

\begin{array} {|c|c|c|c|}
\hline  \ \ T\ \  & \ \ 5\ \  & \ 15\  & \ 30\  \\
\hline M & 36 & 12 & 6 \\
\hline \end{array}

Functions, 2ADV F1 2020 HSC 24

The circle of  `x^2-6x + y^2 + 4y-3 = 0`  is reflected in the `x`-axis.

Sketch the reflected circle, showing the coordinates of the centre and the radius.  (3 marks)

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Show Worked Solution
`x^2-6x + y^2 + 4y-3` `= 0`
`x^2-6x + 9 + y^2 + 4y + 4-16` `= 0`
`(x-3)^2 + (y + 2)^2` `= 16`

 
`=>\  text{Original circle has centre (3, − 2), radius = 4}`

`text(Reflect in)\ xtext(-axis):`

♦ Mean mark 48%.

`text{Centre (3, − 2) → (3, 2)}`
 

Functions, 2ADV F1 EQ-Bank 6

The graph of `f(x)` is shown below. It has asymptotes at  `y = 3`  and  `x = 2`.
 


 

Using interval notation, state the domain and range of `f(x)`.   (2 marks)

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`text(Domain:)\ [−4, 2) ∪ (2, ∞)`

`text(Range:)\ (−∞, ∞)`

Show Worked Solution

`text(Domain:)\ [−4, 2) ∪ (2, ∞)`

`text(Range:)\ (−∞, ∞)`

Functions, 2ADV F1 SM-Bank 36

Consider the function  `f(x) = 1/(x + 2)`.
 

 
 

  1. Sketch the graph  `y = f(−x)`.   (2 marks)

  2. On the same graph, sketch  `y = −f(x)`.   (2 marks)

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i.

ii.   

Show Worked Solution

i.   `text(Sketch)\ \ y = 1/(x + 2)`

`y = f(−x) \ =>\ text(reflect)\ \ y = 1/(x + 2)\ \ text(in the)\ ytext(-axis).`
 

 

ii.   `y = −f(x) \ =>\ text(reflect)\ \ y = 1/(x + 2)\ \ text(in the)\ xtext(-axis).`
 

Functions, 2ADV F1 2019 HSC 13e

  1. Sketch the graph of  `y = |\ x-1\ |`  for  `-4 <= x <= 4`.  (1 mark)

  2. Using the sketch from part i, or otherwise, solve  `|\ x-1\ | = 2x + 4`.  (2 marks)

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  1. `text(See Worked Solutions)`
  2. `(-1, 2)`
Show Worked Solution
i.   

 

ii.    `text(By inspection, intersection when)\ x = -1`

`text(Test:)`

`|-1-1|` `= -2 + 4`
`2` `= 2`

 
`:.\ text(Intersection at)\ (-1, 2)`