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Functions, EXT1 F1 2024 HSC 6 MC

How many real value(s) of \(x\) satisfy the equation

\(\abs{b} = \abs{b\,\sin(4x)}\),

where  \(x \in [0, 2\pi]\) and \(b\) is not zero?

  1. \(1\)
  2. \(2\)
  3. \(4\)
  4. \(8\)
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\(D\)

Show Worked Solution

\(\abs{b} = \abs{b\,\sin(4x)}\ \ \Rightarrow\ \ \abs{1} = \abs{\sin(4x)}\ \ (b \neq 0)\)

\(\sin(4x) = \pm 1\)

\(x \in [0, 2\pi]\ \ \Rightarrow\ \ 4x \in [0, 8\pi]\)

\(\therefore 8\ \text{real solutions}\)

\(\Rightarrow D\)

♦ Mean mark 53%.

Functions, EXT1 F1 EQ-Bank 27

Sketch the graph of  \(y=\abs{x^2-5x+4}\).   (2 marks)

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Show Worked Solution

\(y=\abs{x^2-5 x+4}=\abs{(x-1)(x-4)}\)

\(x\text{-intercepts at} \ (1,0), (4,0)\)

\(y=(x-1)(x-4) \ \text{has low at} \ \ x=\dfrac{5}{2} \ \text{(by symmetry)}\)

\(\Rightarrow \ \text {Low at} \ \left(\dfrac{5}{2},\dfrac{-9}{4}\right)\)
 

Functions, EXT1 F1 2023 HSC 8 MC

The diagram shows the graph of a function.
 

Which of the following is the equation of the function?

  1. \(y=\Big{|}1-\big{|}|x|-2\big{|}\Big{|}\)
  2. \(y=\Big{|}2-\big{|}|x|-1\big{|}\Big{|}\)
  3. \(y=\Big{|}1-\big{|}x-2\big{|}\Big{|}\)
  4. \(y=\Big{|}2-\big{|}x-1\big{|}\Big{|}\)
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\(A\)

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\(\text{By elimination:}\)

\(\text{Even function}\ \rightarrow\ \text{Eliminate}\ C\ \text{and}\ D \)

\(\text{Graph passes through}\ (1, 0) \)

\(\text{Option}\ A:\ \ y=\Big{|}1-\big{|}1-2\big{|}\Big{|} =\Big{|}1-1\Big{|}=0\ \ \text{(lies on graph)} \)

\(\text{Option}\ B:\ \ y=\Big{|}2-\big{|}|1|-1\big{|}\Big{|} =\Big{|}2-0\Big{|}=2\ \ \text{(not on graph)} \)

\(\therefore\ \text{Eliminate}\ B \)

\(\Rightarrow A\)

Functions, EXT1 F1 EQ-Bank 20

Given  `f(x) = x^3-x^2-2x`, without calculus sketch a separate half page graph of the following functions, showing all asymptotes and intercepts.

  1.   `y = |\ f(x)\ |`   (1 mark)

  2.   `y = f(|x|)`   (2 marks)

  3.    `y = 1/(f(x))`   (2 marks)

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a.   

  b.   

c.     

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a.    `f(x)= x^3-x^2-2x= x(x^2-x-2)= x(x-2)(x + 1)`

b.    `y = f(|x|)\ text(is a reflection of)\ y = f(x)\ text(for)\ x > 0\ \text{in the}\ ytext(-axis.)`

c.

Functions, EXT1′ F1 2014 HSC 12a

The diagram shows the graph of a function  `f(x)`.
 

Graphs, EXT2 2014 HSC 12a
 

Draw a separate half-page graph for each of the following, showing all asymptotes and intercepts.

  1.   `y = f(|\ x\ |)`  (2 marks)

  2.   `y = 1/(f(x))`  (2 marks)

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

Graphs, EXT2 2014 HSC 12a Answer1

 

ii.  

Graphs, EXT2 2014 HSC 12a Answer2

Functions, EXT1 F1 2008 HSC 3a

  1.  Sketch the graph of  `y = |\ 2x-1\ |`.   (1 mark)

  2.  Hence, or otherwise, solve  `|\ 2x-1\ | <= |\ x-3\ |`.    (3 marks)

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a.    

Real Functions, EXT1 2008 HSC 3a Answer

b.    `-2 <= x <= 4/3`

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a.    Real Functions, EXT1 2008 HSC 3a Answer

 

b.    `text(Solving for)\ \ |\ 2x-1\ | <= |\ x-3\ |`

`text(Graph shows the statement is TRUE between the)`

`text(points of intersection.)`

`=>\ text(Intersection occurs when)`

`(2x-1)` `= (x-3)\ \ \ text(or)\ \ \ ` `-(2x-1)` `= x-3`
`x` `= -2` `-2x + 1` `= x-3`
    `-3x` `= -4`
    `x` `= 4/3`

 

`:.\ text(Solution is)\ \ {x: -2 <=  x <= 4/3}`