Graphs, MET2 2023 VCAA 1 MC
The amplitude, \(A\), and the period, \(P\), of the function \(f(x)=-\dfrac{1}{2}\sin(3x+2\pi)\) are
- \(A=-\dfrac{1}{2},\ P=\dfrac{\pi}{3}\)
- \(A=-\dfrac{1}{2},\ P=\dfrac{2\pi}{3}\)
- \(A=-\dfrac{1}{2},\ P=\dfrac{3\pi}{2}\)
- \(A=\dfrac{1}{2},\ P=\dfrac{\pi}{3}\)
- \(A=\dfrac{1}{2},\ P=\dfrac{2\pi}{3}\)
Calculus, MET2 2021 VCAA 5
Part of the graph of `f: R to R , \ f(x) = sin (x/2) + cos(2x)` is shown below.
- State the period of `f`. (1 mark)
- State the minimum value of `f`, correct to three decimal places. (1 mark)
- Find the smallest positive value of `h` for which `f(h - x) = f(x)`. (1 mark)
Consider the set of functions of the form `g_a : R to R, \ g_a (x) = sin(x/a) + cos(ax)`, where `a` is a positive integer.
- State the value of `a` such that `g_a (x) = f(x)` for all `x`. (1 mark)
- i. Find an antiderivative of `g_a` in terms of `a`. (1 mark)
- ii. Use a definite integral to show that the area bounded by `g_a` and the `x`-axis over the interval `[0, 2a pi]` is equal above and below the `x`-axis for all values of `a`. (3 marks)
- Explain why the maximum value of `g_a` cannot be greater than 2 for all values of `a` and why the minimum value of `g_a` cannot be less than –2 for all values of `a`. (1 mark)
- Find the greatest possible minimum value of `g_a`. (1 mark)
Graphs, MET2 2021 VCAA 1 MC
The period of the function with rule `y = tan((pix)/2)` is
- `1`
- `2`
- `4`
- `2pi`
- `4pi`
Functions, MET1 2021 VCAA 3
Consider the function `g: R -> R, \ g(x) = 2sin(2x).`
- State the range of `g`. (1 mark)
- State the period of `g`. (1 mark)
- Solve `2 sin(2x) = sqrt3` for `x ∈ R`. (3 marks)
Trigonometry, MET2-NHT 2019 VCAA 2
The wind speed at a weather monitoring station varies according to the function
`v(t) = 20 + 16sin ((pi t)/(14))`
where `v` is the speed of the wind, in kilometres per hour (km/h), and `t` is the time, in minutes, after 9 am.
- What is the amplitude and the period of `v(t)`? (2 marks)
- What are the maximum and minimum wind speeds at the weather monitoring station? (1 mark)
- Find `v(60)`, correct to four decimal places. (1 mark)
- Find the average value of `v(t)` for the first 60 minutes, correct to two decimal places. (2 marks)
A sudden wind change occurs at 10 am. From that point in time, the wind speed varies according to the new function
`v_1(t) = 28 + 18 sin((pi(t - k))/(7))`
where `v_1` is the speed of the wind, in kilometres per hour, `t` is the time, in minutes, after 9 am and `k ∈ R^+`. The wind speed after 9 am is shown in the diagram below.
- Find the smallest value of `k`, correct to four decimal places, such that `v(t)` and `v_1(t)` are equal and are both increasing at 10 am. (2 marks)
- Another possible value of `k` was found to be 31.4358
Using this value of `k`, the weather monitoring station sends a signal when the wind speed is greater than 38 km/h.
i. Find the value of `t` at which a signal is first sent, correct to two decimal places. (1 mark)
ii. Find the proportion of one cycle, to the nearest whole percent, for which `v_1 > 38`. (2 marks)
- Let `f(x) = 20 + 16 sin ((pi x)/(14))` and `g(x) = 28 + 18 sin ((pi(x - k))/(7))`.
The transformation `T([(x),(y)]) = [(a \ \ \ \ 0),(0 \ \ \ \ b)][(x),(y)] + [(c),(d)]` maps the graph of `f` onto the graph of `g`.State the values of `a`, `b`, `c` and `d`, in terms of `k` where appropriate. (3 marks)
Graphs, MET2-NHT 2019 VCAA 2 MC
Calculus, MET2 2019 VCAA 3
During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.
The strength, `f`, of a simple dual-tone frequency signal is given by the function `f(t) = sin((pi t)/3) + sin ((pi t)/6)`, where `t` is a measure of time and `t >= 0`.
Part of the graph of `y = f(t)` is shown below
- State the period of the function. (1 mark)
- Find the values of `t` where `f(t) = 0` for the interval `t in [0, 6]`. (1 mark)
- Find the maximum strength of the dual-tone frequency signal, correct to two decimal places. (1 mark)
- Find the area between the graph of `f` and the horizontal axis for `t in [0, 6]`. (2 marks)
Let `g` be the function obtained by applying the transformation `T` to the function `f`, where
`T([(x), (y)]) = [(a, 0), (0, b)] [(x), (y)] + [(c), (d)]`
and `a, b, c` and `d` are real numbers.
- Find the values of `a, b, c` and `d` given that `int_2^0 g(t)\ dt + int_2^6 g(t)\ dt` has the same area calculated in part d. (2 marks)
- The rectangle bounded by the line `y = k, \ k in R^+`, the horizontal axis, and the lines `x = 0` and `x = 12` has the same area as the area between the graph of `f` and the horizontal axis for one period of the dual-tone frequency signal.
Find the value of `k`. (2 marks)
Graphs, MET2 2019 VCAA 1 MC
Let `f: R -> R,\ \ f(x) = 3 sin ((2x)/5) - 2`.
The period and range of `f` are respectively
- `5 pi` and `[-3, 3]`
- `5 pi` and `[-5, 1]`
- `5 pi` and `[-1, 5]`
- `(5 pi)/2` and `[-5, 1]`
- `(5 pi)/2` and `[-3, 3]`
Graphs, MET2 2018 VCAA 11 MC
The graph of `y = tan(ax)`, where `a ∈ R^+`, has a vertical asymptote `x = 3 pi` and has exactly one `x`-intercept in the region `(0, 3 pi)`.
The value of `a` is
- `1/6`
- `1/3`
- `1/2`
- `1`
- `2`
Algebra, MET2 2018 VCAA 1 MC
Let `f: R -> R,\ f(x) = 4 cos ((2 pi x)/3) + 1`.
The period of this function is
- 1
- 2
- 3
- 4
- 5
Graphs, MET2 2017 VCAA 1 MC
Let `f : R → R, \ f (x) = 5sin(2x) - 1`.
The period and range of this function are respectively
- `π\ text(and)\ [−1, 4]`
- `2π\ text(and)\ [−1, 5]`
- `π\ text(and)\ [−6, 4]`
- `2π\ text(and)\ [−6, 4]`
- `4π\ text(and)\ [−6, 4]`
Graphs, MET1 SM-Bank 27
Algebra, MET2 2010 VCAA 1 MC
The function with rule `f(x) = 4 tan (x/3)` has period
- `pi/3`
- `6 pi`
- `3`
- `3 pi`
- `(2 pi)/3`
Graphs, MET2 2016 VCAA 2 MC
Let `f: R -> R,\ f(x) = 1 - 2 cos ({pi x}/2).`
The period and range of this function are respectively
- `4 and [−2, 2]`
- `4 and [−1, 3]`
- `1 and [−1, 3]`
- `4 pi and [−1, 3]`
- `4 pi and [−2, 2]`
Algebra, MET2 2012 VCAA 1 MC
The function with rule `f(x) = −3sin((pix)/5)` has period
- `3`
- `5`
- `10`
- `pi/5`
- `pi/10`
Functions, MET1 2006 VCAA 4
For the function `f: [– pi, pi] -> R, f(x) = 5 cos (2 (x + pi/3))`
- write down the amplitude and period of the function (2 marks)
- sketch the graph of the function `f` on the set of axes below. Label axes intercepts with their coordinates.
Label endpoints of the graph with their coordinates. (3 marks)
Functions, MET1 2010 VCAA 4a
Write down the amplitude and period of the function
`qquad f: R -> R,\ \ f(x) = 4 sin ((x + pi)/3)`. (2 marks)
Functions, MET1 2011 VCAA 3a
State the range and period of the function
`h: R -> R,\ \ h(x) = 4 + 3 cos ((pi x)/2).` (2 marks)
Algebra, MET2 2014 VCAA 1
The population of wombats in a particular location varies according to the rule `n(t) = 1200 + 400 cos ((pi t)/3)`, where `n` is the number of wombats and `t` is the number of months after 1 March 2013.
- Find the period and amplitude of the function `n`. (2 marks)
- Find the maximum and minimum populations of wombats in this location. (2 marks)
- Find `n(10)`. (1 mark)
- Over the 12 months from 1 March 2013, find the fraction of time when the population of wombats in this location was less than `n(10)`. (2 marks)
Graphs, MET2 2013 VCAA 1
Trigg the gardener is working in a temperature-controlled greenhouse. During a particular 24-hour time interval, the temperature `(Ttext{°C})` is given by `T(t) = 25 + 2 cos ((pi t)/8), \ 0 <= t <= 24`, where `t` is the time in hours from the beginning of the 24-hour time interval.
- State the maximum temperature in the greenhouse and the values of `t` when this occurs. (2 marks)
- State the period of the function `T.` (1 mark)
- Find the smallest value of `t` for which `T = 26.` (2 marks)
- For how many hours during the 24-hour time interval is `T >= 26?` (2 marks)
Trigg is designing a garden that is to be built on flat ground. In his initial plans, he draws the graph of `y = sin(x)` for `0 <= x <= 2 pi` 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 `PC.`
The line through points `P((2 pi)/3, sqrt 3/2)` and `C (c, 0)` is a tangent to the graph of `y = sin (x)` at point `P.`
-
- Find `(dy)/(dx)` when `x = (2 pi)/3.` (1 mark)
- Show that the value of `c` is `sqrt 3 + (2 pi)/3.` (1 mark)
In further planning for the garden, Trigg uses a transformation of the plane defined as a dilation of factor `k` from the `x`-axis and a dilation of factor `m` from the `y`-axis, where `k` and `m` are positive real numbers.
- Let `X prime, P prime` and `C prime` be the image, under this transformation, of the points `X, P` and `C` respectively.
- Find the values of `k` and `m` if `X prime P prime = 10` and `X prime C prime = 30.` (2 marks)
- Find the coordinates of the point `P prime.` (1 mark)
Graphs, MET2 2015 VCAA 1 MC
Let `f: R -> R,\ f(x) = 2sin(3x) - 3.`
The period and range of this function are respectively
- `text(period) = (2 pi)/3 and text(range) = text{[−5, −1]}`
- `text(period) = (2 pi)/3 and text(range) = text{[−2, 2]}`
- `text(period) = pi/3 and text(range) = text{[−1, 5]}`
- `text(period) = 3 pi and text(range) = text{[−1, 5]}`
- `text(period) = 3 pi and text(range) = text{[−2, 2]}`
Algebra, MET2 2013 VCAA 1 MC
The function with rule `f(x) = -3 tan(2 pi x)` has period
- `2/pi`
- `2`
- `1/2`
- `1/4`
- `2 pi`